Handbook, Uncertainty in Gas Turbine Measurements.

Approved for public release; distribution unlimited.

F'n:po j" t:; (, f ~;. ~, !~5. [' F'oresL= ~ l' i.,:' .\," L'

Dr. R. B. Abernethy et al.Pratt & Whitney Aircraft

andJ. W. Thompson, Jr.

ARO, Inc.

February 1973

HANDBOOKUNCERTAINTY IN GAS TURBINE MEASUREMENTS

ENGINE TEST FACILITY

ARNOLD ENGINEERING DEVELOPMENT CENTER

AIR FORCE SYSTEMS COMMAND

ARNOLD AIR FORCE STA TION, TENNESSEE

AEDC-TR-73-5

l'ROPEP:rv OF U.S. AIR FORCE.A.EDC TECI-IN1CAJ.J LffiRARY

ii_P.r-.JOJ r .-4cFB. rt~ 37389

--

-

NOTICBSWhen u. S. Government drawings specifications, or other data are used for any purpose other than adefinitely related Government procurement operation, the Government thereby incurs no responsibilitynor any obligation whatsoever, and the fact that the Government may have formulated, furnished, or inany way supplied the said drawings, specifications, or other data, is not to be regard~d by implicationor otherwise, or in any manner licensing the holder or any other person or corporation, or conveyingany rights or permission to manufacture, use, or sell any patented invention that may in any way berelated thereto.

Qualified users may obtain copies of this report from the Defense Documentation Center.

References to named commercial products in this report are not to be considered in any sense as anendorsement of the product by the United States Air Force or the Government.

HANDBOOKUNCERTAINTY IN GAS TURBINE MEASUREMENTS

Dr. R. B. Abernethy et al.Pratt & Wh itney Aircraft

and

J. W. Thompson, Jr.ARO, Inc.


AEDC-TR-73-5

RASCOE-FH

Typewritten Text

AEDC Technical Library Arnold AFB, TN, 37389

AEDC-TR-73-5

FOREWORD

The work reported herein was sponsored by the Arnold Engineering DevelopmentCenter, Air Force Systems Command, United States Air Force, under Program Element65802F.

The results presented were compiled by· ARO, Inc. (a subsidiary of Sverdrup &Parcel and Associates, Inc.), contract operator of the Arnold Engineering DevelopmentCenter (AEDC) , Air Force Systems Command (AFSC), Arnold Air Force. Station,Tennessee, under Contract F40600-73-C-0004. The preparation of the text wasaccomplished by Dr. R. B. Abernethy, Senior Project Engineer, Billy D. Powell, David L.Colbert, and Daniel G. Sanders, Pratt & Whitney Aircraft, under subcontract to ARO, Inc.The contracted work consisted of a revision to the material in the "Interagency ChemicalRocket Propulsion Group (ICRPG) Handbook for Estimating the Uncertainty inMeasurements made with Liquid Propellant Rocket Engine Systems," CPIA PublicationNo. 180 (same authors as above), substituting treatment of gas turbine measurementerrors for rocket engine treatment and writing additional material applicable to gasturbine measurement errors. The report was prepared under ARO Project No. RW5245,and the manuscript was submitted for publication on May 8, 1972.

The authors are indebted to the many engineers and statisticians who havecontributed to the work. A few must be noted for their particular contributions, Dr.Joan Rosenblatt, Dr. H. H. Ku, and J. M. Cameron of the National Bureau of Standardsfor their helpful discussions and comments on both this handbook and CPIA 180, andsimilarly, R. E. Smith, Jr., Chief of T-Cells Division, as well as T. C. Austin, C. R.Bartlett, W.O. Boals, Jr., and T. J. Gillard of ARO, Inc., at the Arnold EngineeringDevelopment Center. Engineers at Pratt & Whitney Aircraft, Florida and Connecticutfacilities, provided the authors with constructive and spirited criticism in every section.Various technical committees under the American Society of Mechanical Engineers(ASME), the American Institute of Aeronautics and Astronautics, and the InternationalStandards Organizations expressed interest and comments.

This technical report has been reviewed and is approved.

EULES L. HIVELYResearch and Development DivisionDirectorate of Technology

ii

ROBERT 0. DIETZDirector of Technology

ABSTRACT

The lack of a standard method for estimating the errors associated with gas turbineperformance data has made it impossible to compare measurement systems betweenfacilities, and there has been confusion over the interpretation of error analysis.Therefore, a standard uncertainty methodology is proposed in this Handbook. Themathematical uncertainty model l presented is based on two components of measurementerror: the fixed (bias) error and the random (precision) error. The result of applying themodel is an estimate of the error in the measured performance parameter. Theuncertainty estimate is the interval about the measurement which is expected toencompass the true value. The propagation of error from basic measurements throughcalculated performance parameters is presented. Traceability of measurement back to theNational Bureau of Standards and associated error sources is reviewed.

iii

AEDC-TR-73-5

CONTENTS

"/

r./n.

III.

ABSTRACT .,.INTRODUCTION1.1 Objective . .1.2 Scope ....1.3 Measurement Error

1.3.1 Precision (Random Error)1.3.2 Bias (Fixed Error)

1.3.2.1 Large Known Biases1.3.2.2 Small Known Biases1.3.2.3 Large Unknown Biases1.3.2.4 Small Biases, Unknown Sign, and Unknown Magnitude1.3.2.5 Small Biases, Known Sign, and Unknown Magnitude

1.4 Measurement Uncertainty . . . . . .1.5 Propagation of Measurement Errors .

1.5.1 Engine Inlet Airflow .. . . . . . . . . . .1.5.2 Thrust Specific Fuel Consumption (TSFC) .

1.6 Measurement Process1.7 Reporting Error . . . . .1.8 TraceabilityUNCERTAINTY MODEL2.1 General . . . . . . .2.2 Measurement Error Sources

2.2.1 Calibration Hierarchy Errors2.2.2 Data Acquisition Errors2.2.3 Data Reduction Errors .

2.3 Measurement Uncertainty Model2.4 Example of the Model .....

2.4.1 Net Thrust Measurement2.4.2 Fuel Flow Measurement2.4.3 Thrust Specific Fuel Consumption

2.5 Summary . . . . . .FORCE MEASUREMENT . . . . . . .3.1 General . . . . . . . . . . . . . .3.2 Force Measurement Error Sources

3.2.1 Force Transducer Calibration Hierarchy3.2.1.1 Precision Index3.2.1.2 Degrees of Freedom3.2.1.3 Bias .3.2.1.4 Uncertainty .

3.2.2 Data Acquisition and Reduction Errors3.2.2.1 Applied Load Tests .....3.2.2.2 Elemental Error Evaluation ..

3.2.2.2.1 Stand Mechanics

v

iii11112233334489

10131314171718181919202224242526292930303233333435363839

III.

IV.

AEDC-TR-73-5

FORCE MEASUREMENT (Continued)3.2.2.2.2 Fuel Line Temperature Variations3.2.2.2.3 Fuel Line Pressure Variations3.2.2.2.4 Force Transducer Temperature and

Ambient Pressure Variations .3.2.2.2.5 Excitation Voltage Errors .3.2.2.2.6 Recording System Electrical Calibration3.2.2.2.7 Analog-to..Digital Conversion Errors3.2.2.2.8 Recording System Resolution.3.2.2.2.9 Electrical Noise .3.2.2.2.10 Tare Variations .3.2.2.2.11 Computer Resolution

3.3 Force Measurement Error Analysis3.4 End-to..end Calibration . .3.5 Summary .FUEL FLOW MEASUREMENT4.1 General . . . . . . . . . .4.2 Fuel Flow Measurement Error Sources

4.2.1 Calibration Errors . . . . . . .4.2.1.1 Volumetric Calibration.

4.2.1.1.1 Calibration of the Interlab Standard4.2.1.1.2 Uncertainty in the Working Standard.

4.2.1.2 Gravimetric Calibration4.2.1.3 Calibration by Comparison.

4.2.2 Data Acquisition Errors4.2.2.1 Multiple Instruments

4.2.3 Data Reduction Errors . . . ..4.2.3.1 Density Determination Errors4.2.3.2 Computer Resolution

4.3 Fuel Flow Measurement Errors4.4 End-to-end Calibration . . . . . . . . .4.5 Summary . . . . . . . . . . . . . . . .

V. PRESSURE AND TEMPERATURE MEASUREMENTS5.1 General . . . . . . . . . . . . . . .5.2 Pressure Measurement Error Sources .

5.2.1 Calibration Hierarchy Errors .5.2.2 Data Acquisition and Reduction Errors5.2.3 Probe Errors .5.2.4 Pressure Measurement Error Summary

5.3 Temperature Measurement Error Sources5.3.1 Calibration Hierarchy Errors .5.3.2 Data Acquisition and Reduction Errors

5.3.2.1 Thermocouples .5.3.2.2 Resistance Thermometers .

5.3.3 Temperature Measurement Error Summary5.3.3.1 Thermocouples .....5.3.3.2 Resistance Thermometers ....

vi

4142

42444648494950505154555757585859596164666870717172737376777779798385878888929395979797

AEDC-TR-73-5

VI. AIRFLOW . . . . . . . . . . . . . . . . . . 996.1 General . . . . . . . . . . . . . . . . . 996.2 Airflow Rate Measurement Techniques . 100

6.2.1 Subsonic F10wmeters . . . . . . 1006.2.1.1 Venturis and Nozzles 1006.2.1.2 Orifices ..... 105

6.2.2 Critical Venturi Flowmeters ... 1066.2.3 Calibration Techniques . . . . . 108

6.2.3.1 Calibration by Calculation 1086.2.3.2 Experimental Calibration 1086.2.3.3 Calibration by Fabrication 109

6.3 Elemental Error Sources .. . . 1096.3.1 Discharge Coefficient . . . . . . . . 109

6.3.1.1 Calculated Cd 1096.3.1.2 Experimentally Determined Cd 110

6.3.1.2.1 Comparison with a Standard Flowmeter 1106.3.1.2.2 Calibration by Traverse. . 1126.3.1.2.3 Calibration by Liquid 1156.3.1.2.4 Calibration by Fabrication . . . . . . 116

6.3.2 Non-Ideal Gas Behavior and Variation in Gas Compositions . 1166.3.3 Thermal Expansion Correction Factor . . . . . . . 1166.3.4 Ratio of Specific Heats and Compressibility Factor 11 76.3.5 Measurement Systems 117

6.4 Propagation of Error to Airflow 1176.4.1 Critical-Flow Venturi 1176.4.2 Subsonic Orifice 119

VII. NET THRUST AND NET THRUST SPECIFIC FUEL CONSUMPTION 1217.1 General . . . . . . . . . . . . . . . . . 1217.2 Gross Thrust Measurement Techniques 121

7.2.1 Scale Force Method 1217.2.2 Momentum Balance Method .. 122

7.3 Propagation of Errors to Net Thrust . . 1247.4 Propagation of Error to Net Thrust Specific Fuel Consumption . 128

VIII. SPECIAL METHODS. . . . . . . . . . . . . . . . . . . . . . . . . . 1298.1 General . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1298.2 Measurement Uncertainty for Multi-Engine Installations (Similar Engines) 129

8.2.1 General . . . . . . . . . . . . . . . . 1298.2.2 Example of a Four-Engine Installation 130

8.3 Measurement Processes . . . . . . . . . . 1308.3.1 Many Stand, -Many Engine Model 1318.3.2 Single Stand, Single Engine Model 131

8.4 Confidence Interval when Biases are Negligible or can be Ignored . 1328.5 Compressor Efficiency Error Analysis 133

8.5.1 General . . . . . . . . . . . . . . . . . . . . . . . . . . . 133

vii

AEDC-TR-73-5

VIII. SPECIAL METHODS (Continued)8.5.2 The General Process . . . . . . . . . . . 1338.5.3 Single Stand, Single Compressor Process 135

8.6 How to Interpret Uncertainty 1368.7 Dynamic Measureluent Uncertainty 137

IX., GLOSSARY . . . . . . . . . . . . . . . 139APPENDIXES

A. Precision Index for Uniform Distribution of Error 147B. Propagation of Errors by Taylor's Series . . . . . . 149C. Estimates of the Precision Index from Multiple Measurements 157D. Outlier Detection 163E. Tables.............................. 167

Figure

I-I1-21-31-41-51-61-71-8

II-I11-211-311-411-5

11-6111-1111-2111-3111-4111-5111-6111-7111-8111-9111-10111-11

111-12

ILLUSTRATIONS

Measurement ErrorPrecision ErrorBias ErrorFive Types of Bias ErrorMeasurement Error (Bias, Precision, and Accuracy)Measurement Uncertainty, Symmetrical Bias .' .Measurement Uncertainty, Nonsymmetrical BiasFlow through a Choked Venturi .....Force Measurement Calibration HierarchyData Acquisition System .Calibration Curve . . . . . . . . . . . . .Uncertainty Parameters, U = ±(B + t95 S)Overall Uncertainty Model

a. General View . . . .b. Propagation of Errorc. Elemental Errors

- -

Logic Decision DiagramForce Measurement SystemForce Transducer Calibration HierarchyCalibration Curves .Scatter in Measured Force .Calibration Hierarchy Elemental BiasCalibration Process Uncertainty Parameter, U1 = ±(B1 + t95 Sl)Gas Turbine Thrust Measurement System Calibration Configuration .Precision Errors . . . . . . . . . . . . . . . . . .Temperature Bias Effect on Distribution of ErrorsAmbient Pressure Effect on Load Cell OutputErrors

a. Precision Error .b. Bias Error .c. Both Bias and Precision Errors

Elemental Precision Error of Calibration Power Supply

viii

12225679

18191922

2323232729303132333439404343

45454547

Figure

111-13111-14111-15

111-16

IV-lIV-2IV-3IV-4IV-5IV-6IV-7IV-8IV-9IV-IOV-IV-2V-3V-4

V-5

V-6V-7V-8V-9V-IOV-IIV-12

VI-lVI-2VI-3

VI-4VI-5VI-6VI-7VI-8VI-9

VII-l

VII-2

Sensitivity Curve .Tare History Showing Elemental Precision ErrorTypical Calibration Data from Force Measuring SystemUsed in Engine Sea-Level Testing .Typical Calibration Data from Force Measuring SystemsUsed in Engine Altitude TestingTurbine Meter Signal .Turbine Meter Calibration Curve . . . . . . . . .Turbine Meter Volumetric Calibration HierarchyVolumetric Calibrator .Turbine Meter Gravimetric Calibration HierarchyGravimetric Calibrator .. . . . . . . .Turbine Meter Comparison Calibration HierarchyComparative Calibration .Data Acquisition System CalibrationEnd-to-End Calibration . . . . . . .Strain-Gage Pressure Transducer CircuitryPressure Transducer Calibration with Pressure StandardDeadweight Piston Gage .Temperature Measurement

a. Resistance Thermometer Three-Wire Systemb. Thermocouple System

Probe Boss Arrangementa. Front View .b. Side View .

Pressure Transducer Calibration HierarchyDeadweight Piston Gage CalibrationPrecision Index at Any Applied Pressure (P)Periodic Pressure Tests .Temperature Transducer Calibration HierarchyTypical Thermocouple Channel .Temperature Data Recording CalibrationSchematic of Typical Venturi and Nozzle with Measuring StationsSchematic of Typical Orifice with Measuring StationsSchematic of Critical Venturi Flowmeter Installation Upstreamof a Turbine Engine .Discharge Coefficient Error DistributionCalibration by ComparisonFlowmeter Throat TraverseFlat. Mass-Velocity ProfileDistorted Mass-Velocity ProfileShaded Area Calculated as a Function of dl and d2Freebody Diagram for External Forces (Scale Force)Method of Determining Engine Gross (Jet) ThrustFreebody Diagram for Internal Forces (Momentum Balance)Method of Determining Engine Gross (Jet) Thrust .....

ix

AE DC-TR-73-5

4850

55

5557585961646566666874777778

7878

797979798384889395

100105

106109110112113113114

122

123

AEDC-TR-73-5

Figure

VIII-lVIII-2

IX-lIX-2

I.II.

III.IV.V.

VI.VII.

VIII.IX.X.

XI.XII.

XIII.XIV.XV.

XVI.XVII.

XVIII.

XIX.XX.

XXI.XXII.

XXIII.XXIV.

~easurement lJncertaintyRun-to-Run DifferencesBias in a Random ProcessCorrelation Coefficients

TABLES

Nonsymmetrical Bias Limits .lJncertainty Intervals Defined by Nonsymmetrical Bias Linnits .Flow Data . .lJncertainty Components .Calibration Hierarchy Error SourcesData Acquisition Error SourcesData Reduction Error SourcesCalibration Hierarchy Error SourcesCalibration Data, .Data Acquisition Error SourcesData Reduction Error SourcesForce ~easurement Elemental Error ValuesElemental Errors .International Practical Temperature Scale of 1968Elemental Error for Calibration by ComparisonAirflow ~easurement Error Source .Airflow Error Source .... . . . . . . . . . . . .Typical ~easurement and lJncertainty Values Qsed in Net Thrustfor Supersonic Afterburning Turbofan Engine .Derived ~easurement lJncertainty ValuesA ~easurement System with Six Error SourcesTabulation of the Elemental ErrorsSummary of Errors . . . . . . . . . .Elemental Errors .lJncertainty Values for Two Processes

x

137137139140

47

101318191930313535517689

111118119

127127131133134136136

AE DC-T R-73-5

SECTION IINTRODUCTION

1.1 OBJECTIVE

The objective of this Handbook is to present a standard method of treatingmeasurement error! or uncertainty for gas turbine engine performance parameters, suchas thrust, airflow, and thrust specific fuel consumption. The need for a standard isobvious to those who have reviewed the numerous methods currently used. The subject iscomplex and involves both engineering and statistics. Only one method is presentedherein without alternative paths. A single standard method is required to makecomparisons between engine manufacturers and between facilities. However, it must berecognized that no single method will give a rigorous, scientifically correct answer for allsituations. Further, even for a single set of data, the task of finding and proving onemethod to be correct is usually impossible. The method selected is believed to be mostuniversally applicable. It is identical with the measurement uncertainty model used in therocket engine industry which has been well received ("ICRPG Handbook for Estimatingthe Uncertainty in Measurements made with Liquid Propellant Rocket Engine Systems,"CPIA No. 180, AD855l30, April 30, 1969).

There are numerous examples for illustration. An effort has been made to use simpleprose with a minimum of jargon.

1.2 SCOPE

This Handbook presents a working outline detailing and illustrating the techmquesfor estimating measurement uncertainty. Section II describes the mathematical model fora typical performance parameter (thrust specific fuel consumption). Sections III, IV, V,and VI treat errors associated with the measurement of force, fuel flow, pressure andtemperature, and airflow. Each section includes a discussion of the methods of calibrationand lists of the elemental errors and examples of the statistical techniques. Section VIIdescribes the calculations of the uncertainty in net thrust and thrust specific fuelconsumption at altitude conditions. Section VIII describes and illustrates several specialmethods. Section IX is the Glossary. Appendixes with tables, derivations, and proofs arefound at the end of the Handbook.

1.3 MEASUREMENT ERROR

I

1.00.995

True (NBS) Value

Error

0.990

Fig. 1-1 Measurement Error

0.985

Parameter Measurement Value

Measured Value

I

0.980

All measurements have measurementerrors. These errors are the differencesbetween the measurements and the truevalue defined by the National Bureau ofStandards (NBS). Uncertainty is the maximum error which might reasonably be

----4--..::---I-------+----+-~~~

expected and is a measure of accuracy, Le.,the closeness of the measurement to the truevalue. Measurement error has two components: a fixed error and a random error.

! For a deftnition of terms used in this Handbook, see the Glossary in Section IX.

1

AEDC-TR-73-5

1.3.1 Precision (Random Error)

N - 1

N _ 2:£ (X. - X)i=l 1

s

Standard DeviationEstimate of a

Scatter Due toPrecision Error

1.015

Average Measurement

1.00.985

'Hoi»C)

l:lQ)

50Q)

HI'<i '--__....::::LJL.LU..LL.LLL.l..u.LL ---::::::-. _

Random error is seen in repeated measurements. Measurements do not and are notexpected to agree exactly. There are always numerous small effects which causedisagreements. The variation between repeated measurements is called precision error. The

standard deviation (a) is used as ameasure of the precision error. Alarge standard deviation meanslarge scatter in the measurements.The statistic (s) is calculated toestimate the standard deviationand is called the precision index

Par4meter Measurement Value

Fig. 1-2 Precision Error where N is the number of measurements made and X is the averagevalue of individual measurements Xi.

1.3.2 Bias (Fixed Error)

The second component, bias, is theconstant or systematic error. In repeatedmeasurements, each measurement has thesame bias. The bias cannot be determinedunless the measurements are comparedwith the true value of the quantitymeasured.

Bias is categorized into five classes:(l) large known biases, (2) small known

biases, (3) large unknown biases, andsmall unknown biases which may have(4) unknown sign (±) or (5) known sign.

;True (NBS) ValueAverage Measurement

I-"e---Btas-.-..-t

Parameter Measurement Value

Fig. 1-3 Bias ErrorKnown Sign

pnknown Magnitudeand Magnitude

Large(1) Calibrated (3) Assumed to be

Out Eliminated

(2) Negligible, (4) Unknown I (5) KnownContributes Sign Sign

Small toBias Limit Contributes to

Bias Limit

Fig. 1-4 Five Types of Bias Errors

2

AEDC-TR-73-5

1.3.2.1 Large Known Biases

The large known biases are eliminated by comparing the instrument with a standardinstrument and obtaining a correction. This process is called calibration.

1.3.2.2 Small Known Biases

Small known biases mayor may not be corrected depending on the difficulty of thecorrection and the magnitude of'the bias.

1.3.2.3 Large Unknown Biases

Unknown biases are not correctable. That is, they may exist, but the magnitude of thebias is not known, and perhaps even the sign is not known.

Every effort must be made to eliminate all large unknown biases. The introductionof such errors converts the controlled measurement process into an uncontrolledworthless effort. Large unknown biases usually come from human errors in dataprocessing, incorrect handling and installation of instrumentation, and unexpectedenvironmental disturbances such as shock and bad flow profiles. In a well-controlledmeasurement process, the assumption is that there are no large unknown biases. To ensurethat a controlled measurement process exists, all measurements should be monitored withstatistical quality control charts. A list of references describing the use of statisticalquality control charts is included at the end· of this section. Drifts, trends, andmovements leading to out-of-control situations should be identified and investigated.Histories of data from calibrations are required for effective control. It is assumedthroughout this Handbook that these precautions are observed and that the measurementprocess is in control; if not, the methods contained herein are invalid.

1.3.2.4 Small Biases, Unknown Sign, and Unknown Magnitude

In most cases, the bias error is equally likely to be plus or minus about themeasurement. That is, it is not known if the limit is positive or negative, and the estimatereflects this. The bias limit is estimated as an upper limit on the maximum fixed error.For example, ±5 pounds IS a typical bias limit.

It is both difficult and frustrating to estimate the limit of an unknown bias. Todetermine the exact bias in a measurement, it would be necessary to compare the truevalue and the measurements. This is almost always impossible. An effort must be made toobtain special tests or data that will provide bias information. The following are examplesof such data:

1. Interlab, interfacility, intercompany tests on measurement devices, testrigs, and full-scale engines.

2. Flight test data versus altitude test chamber data versus ground test data.

3. Special comparisons of standards with instruments in the actual testenvironment.

3

AEDC·TR·73-5

4. Ancillary or concomitant functions that provide the same performanceparameter; i.e., in an altitude engine test, airflow may be measured with(l) an orifice and (2) a bellmouth, (3) estimated from compressorspeed-flow rig data, (4) estimated from turbine flow parameter, and (5) jetnozzle calibrations.

5. When it is known that a bias results from a particular cause, specialcalibrations may be performed allowing the cause to perturbate through itscomplete range to determine the range of bias.

If there is no source of data for bias, the judgment of the most knowledgeableinstrumentation expert on the measurement must be used. However, without data, theupper limit on the largest possible bias error must reflect the lack of knowledge.

1 Small Known Sign, and Unknown Magnitude

Sometimes the physics of the measurement system provide knowledge of. the signbut not the magnitude of the bias. For example, thermocouples radiate and conductenergy to indicate lower temperatures. The bias limits which result are nonsymmetrical,Le., not of the form ±b. They are of the form ~g where both limits may be positive ornegative or the limits may be of mixed sign as indicated. Table I below lists severalnonsymmetrical bias limits for illustration.

Table I Nonsymmetrical Bias limits

Bias Limits Explanation

0, +10 deg The bias will range from zero to plus 10 deg.-5, +15lb The bias will range from minus 5 to plus 15 lb.

+3, +7 psia The bias will range from plus 3 to plus 7 psia.-8, -3 deg The bias will Tange from minus 8 to minus 3 deg.

In summary, measurement systems are subject to two types of errors, bias andprecision error (Fig. 1-5). One sample standard deviation is used as the precision index.The bias limit is estimated as an upper limit on the maximum fixed error.

1 MEASUREMENT UNCERTAINTY

For simplicity of presentation a single number (some combination of bias andprecision) is needed to express a reasonable limit for error. The single number must havea simple interpretation (the largest error reasonably expected) and be useful withoutcomplex explanation. It is impossible to define a single rigorous statistic because the bias isan upper limit based on judgment which has unknown characteristics. Any function of thesetwo numbers must be a hybrid combination of an unknown quantity (bias) and a statistic(precision). However, the need for a single number to measure error is so great that theadoption of an arbitrary standard is warranted. The standard most widely used is

4

True Value andAverage of AllMeasurements

AEDC-TR-73-5

Average of AllMeasurements

>.t)s::(1)g. True Value(1)

1-lrz.t

Parameter Measurement

a. Unbiased, Precise, Accurate

. True Value andAverage of AllMeasurements'


c. Unbiased, Imprecise, Inaccurate


b. Biased, Precise, Inaccurate

Average of AllMeasurements

>.t)

~ Trueg.(1)

1-lrz.t


d. Biased, Imprecise, Inaccurate

Fig. 1-5 Measurement Error (Bias, Precision, and Accuracy)

the bias limit plus a multiple of the preCISIon index. This method is recognized andrecommended by the NBS2 and has been widely used in industry.

Uncertainty (Fig. 1-6) may be centered about the measurement and is defined herein as:

(1-1)

where B is the bias limit, S is the precision index, and t95 is the 95th percentile pointfor the two-tailed Students "t" distribution (Table E-l, Appendix E). The t value is afunction of the number of degrees of freedom (df) used in calculating S. For smallsamples, t will be large, and for larger samples t will be smaller, approaching 1.96 as alower limit. The use of the t arbitrarily inflates the limit U to reduce the risk ofunderestimating S when a small sample IS used to calculate S. Since 30 degrees of freedomyield a t of 2.04 and infinite degrees of freedom yield a t of 1.96, an arbitrary selectionof t =2 for values of df from 30 to infinity was made, i.e., U =±(B + 2S), when df ~ 30.

In '!~~~~!!1pl~.,.Jl1~ nll~~~r.of d~gr~es()Lfreedomis.the siz~ Qfthe saIllple. When astaifstiC--is calculated from the sainple, the· degrees of freedom associated with the statistic

2Eisenhart, C. "Expression of Uncertainties of Final Results, Precision Measurement and Calibration," NBSHandbook 91, Vol I, February 1969, pp. 69-72.

Ku, H. H. "Expressions of Imprecision, Systematic Error, and Uncertainty Associated with a Reported Value,Precision Measurement and Calibration," NBS Handbook 91, Vol I, February 1969, pp. 73-78.

5

AEDC-TR-73-5

are reduced by one for every estimated parameter used in calculating the statistic. Forexample, from a sample of size N, X is calculated:

which has N degrees of freedom and

_ N

X = S X·/Ni= 1 1

(1-2)

s =

N _ 2

L (x. - x)i=l 1

N-l

(1-3)

which has N-l degrees of freedom because :x (based on the same sample of data) is usedto calculate S. In calculating other statistics, more than one degree of freedom may belost. For example, in calculating the standard error of a curve fit, the number of degreesof freedom which are lost is equal to the number of estimated coefficients for the curve.

It is recommended that the uncertainty parameter (D) be used for simplicity ofpresentation; however, the components (bias, precision, and degrees of freedom) shouldbe available in an appendix or in supporting documentation. These three componentsmay be required (l) to substantiate and explain the uncertainty value, (2) to provide asound technical base for improved measurements, and (3) to propagate the uncertaintyfrom measured parameters to performance parameters, and from performance parameters

Measurement

~---Largest Negative Error- (B + t 95S)

Largest Positive Error----~

+(B + t 95S)

....--- -B

Measurement Scale

Range of ±t95S................................. Precision ----.........-

Error+B

.....--------Uncertainty Interval-----------~

(The True Value Should Fall within This Interval)

Fig. 1-6 Measurement Uncertainty, Symmetrical Bias

6

AEDC-TR-73-5

to other more complex performance parameters (i.e. fuel flow to Thrust Specific FuelConsumption (TSFC), TSFC to aircraft range, etc.). Nthougl1uncertainty is.uQt .. astatisticll.LconfideIlce interval, it is an arbitrary substitute which is probably bestin!erpre!edas the largest error expecteq .. Under any reasonable assumption for thedlsfributlon' of bias, the coverage of U is greater than 95 percent, but this cannot beproved as the distribution of bias is both unknown and unknowable.

If there is a nonsymmetrical bias limit (Fig. 1-7), the uncertainty U is no longersymmetrical about the measurement. The upper limit of the interval is defined by theupper limit of the bias interval (B+). The lower limit is defined by the lower limit of thebias interval (B-).

Measurement

......-- Largest Negative Error

(B- - t95S)

LargestPositiveError

+(B + t95S)

Measurement Scale

±t95SRange of

,......------B-----_+-----Precision---...t..... B+Error

......---------tJncertainty I nterval-------.--,.....(The True Value Should Fall within This Interval)

Fig. 1-7 Measurement Uncertainty, Nonsymmetrical Bias

The uncertainty interval U is U- = B- - t95 S to U+ = B+ + t95 S.

Table II shows the undertainty U for the nonsymmetrical bias limits of Table I. TheSand t95 are assumed to be 1 unit and 2 units for each case.

Table II Uncertainty Intervals Defined by Nonsymmetrical Bias Limits

B- B+ t95 S U- U+(Lower limit for U) (Upper limit for U)

odeg +10 deg 2 deg -2 deg +12 deg-SIb +151b 2lb -71b +171b+3 psia +7 psia 2 psia +1 psia +9 psia-8 deg -3 deg 2 deg -10 deg -1 deg

7

AEDC-TR-73-5

1'l1~ proper method for combining elemental measurement uncertaintyyalues is t()d~t~~~net~e root-sum-square values of the elemental bias limits and th~el~tl1entalprecision indices separately. Thell~ ~pply the uncertainty formula to the c?l11bill~~ ~hlSlimits and precision indices. In-s-ome cases, the same value will be obtained it -the

--uncertainties are root-sum-squared directly. However, this is not a general rule, and largeerrors in the combined uncertainty (lO to 25 percent) can result. Further, theroot-sum-squared uncertainty value will be smaller (optimistic) than the properuncertainty estimate, and the estimate is a significant underestimate of the truemeasurement error.

For example, in combining the following uncertainties the root-sum-square of theuncertainties was 18.38 units. The correct value was 23.21 units.

Bias Limit (B) Precision Index (8) Uncertainty

1 6 ±1311 1 ±13

where Uncertainty = ±(B + 28).

Now the bias limit for the combined parameter is the root-sum-square of 1 and 11:

The precision index for the combined parameter is the root-sum-square of 6 and 1:

5 = ..J 6 2 + 12= Vii = 6.08

I

The Uncertainty is thus:

U = ±(B + 25) = ±[11.05 + 2 (6.08)]

The root-sum-square of the original uncertainties is

±23.21

Now,

~ (13) 2 + (13) 2= ~ 169 + 169 = V338 18.38

23.21 - 18.38 x 100 = 26.3%18.38

and over 25 percent enur nas been introduced just because of the wrong propagation oferror formula.

1.5 PROPAGATION OF MEASUREMENT ERRORS

Rarely are performance parameters measured directly; usually more basic quantitiessuch as temperature, force, pressure, and fuel flow are measured, and the performanceparameter is calculated as a function of the measurements. Error in the measurements ispropagated to the parameter through the function. The effect of the propagation may beapproximated with the Taylor's series methods.

8

AEDC-TR-73-5

1.5.1 Engine Inlet Airflow

Engine inlet airflow is determined by the use of a chokedventuri and measurements of upstream temperature and stagnation pressure (Fig. 1-8).

The flow (Wa) is calculated from

where

Flow

Engine Inlet Airflow 'Measurement, Wa

-------Wa

VenturiThroat, At

Fig. 1-8 Flow through a Choked Venturi

ToEngineInlet

FA is the factor to account for thermal expansion of the venturiAeff is the effective venturi throat areaPI is the total (stagnation) pressure upstreamTI is the total temperature upstreamC* is the factor to account for the properties of the air

(critical flow constant)

The precision index for the flow (Sw a) is calculated using the Taylor's seriesexpansion (this method is derived in Appendix B):

where

(aw )2 (aW)2a S 6+ -- c* + -- Sa~ * \ aA e fi A e ££ (

aw )2+ _',_6 Sp

<1P 11

(aw )2+ _6 ST (1-5)aT 1 1

awa

-a denotes the partial derivative of Wa with respect to FA·FA

Taking the necessary derivatives gives

S _ I(c. AeffP1 )2 (F aAefll 12(F aC·Pl )2 (F aC· Aeff )2 (F aC· AeffP1 )2

W - SF + Sc + --- SA + Sp + STa v"'f; A ~ ~ e ff ' ~ 1 , -2N 1

1 (1-6)(

By inserting the nominal values and preCISIon errorsllfrom Table III into Eq. (1-6), theprecision index of 0.3658 lb/sec for engine airflow is obtained.

The bias limit in the flow calculation is propagated from the bias limits of themeasured variables. Using the Taylor's series formula gives

(1-7)

9

For this example, where Wa = FA C*AeffPl/VT1:

BW =a

Taking the necessary derivatives gives

BW =a (

F A P ~2+ a eff 1 B * ,VTl C

(1-9)

By inserting the nominal values and bias limits of the measured parameters fromTable III into Eq. (1-9), a bias limit of 0.6987 lb/sec is obtained for a nominal engineairflow of 248.23 lb/sec.

Table III Flow Data

Wa = FA C*Aeff P1 Iv'f; .

Parameter Units Nominal Precision Index Bias Limit(One Standard Deviation)

FA . --- 1.00 0.0 0.001

C*Ibm R%

0.532 0.0 0.000532Ib sec

. Aeff in.2 296. 0.148 0.592PI psia 36.8 0.05 0.05TI OR 545. 0.3 0.3

::WaIttm 248.23 0.3658 0.6987sec

To, .propagate nonsymmetrical bias limits, the bias limit portion of the analysis must?e ... co1l1pleted for both the upper and the lower limits. Then, the two results arecom~rned as illu~trated in Table II. There is a more detailed illustration of propagation ofnonsymmetrical bias limits in Section VIII.

1.5.2 Thrust Specific Fuel Consumption (TSFC)

The goal of any analysis of measurement system errors is to determine the resultingerrors in the reduced parameters, for example TSFC, which is calculated as the ratio offuel flow (Wr) to net thrust (FN); TSFC = Wr/FN. Net thrust and TSFC uncertaintycalculations are described in Section VII. The technique for relating the errors ofmeasurement to the errors in the reduced parameters is based on a Taylor's Seriesexpansion from the calculus. The Taylor's expression for errors in thrust specific fuelconsumption is

10

~TSFC = aTSFC ~W aTSFC ~F = _1 ~Wf _ ~ ~FaW f + aF N F F 2 N

f N N N

AEDC-TR-73-5

(1-10)

Where aTSFC/aWr and aTSFC/aFN are the partial derivatives of thrust specific fuelconsumption with respect to fuel flow and net thrust. The precison index isapproximated by

(1-11 )

For example, the following hypothetical data were used to estimate thrust specific fuelconsumption uncertainty:

Bias Precision Degrees of UncertaintyParameter Nominal Limit Index Freedom Limit

Thrust (FN) 10,000 18.1 lbf 37.81bf 57 93.7 1bfFuel Flow (Wr) 10,000 50lbm/hr 501bm/hr 60 150lbm/hr

The nominal thrust specific fuel consumption is calculated from Wr!FN :

Wf _ 10,000 Ibm/ h r

F N - 10,000 1b f1.0 Ibm/lbf-hr

The precision index of thrust specific fuel consumption is

(50 \2 (-10,000 X 37.~2

10,000J + 10,0002 >J

= 1O.00631bm/lbf-hr)

The propagation formula is similar for bias

( ~ 2 ( ~2~ + -10,000 18 110,000 10,0002'

(1-12)

(1-13)

= 10.0053 Ibmllbf-hr

11

;

)

AEDC-TR-73-5

'f~e~~~eesof fre~~lo-mf()rtl1e l'~FC preClSlOn index can be foun~ using theWelch~Satterthwaite technique. In--this situation, the partial derivative weighting factors,which are used in the calculation of the precision index, must also be used in theWelch-Satterthwaite formula. Note: The calculation is carried. out to illustrate the use ofthe partial derivatives with t~--W~l~h-Satterthwaite.It is not necessary to c(ll~~l~te-tfiedegrees of freedom for TSFC since the degrees of freedom for thrust ancL[uel flgw are57 and 60, respectively. The expected minimum result would be 57. The t multiple_isessentially 2.0 for degrees of freedom greater than thirty (Section 1.4). When the degreesof freedom for each component are greater than 30, the Welch-Satterthwaite procedure-can be omitted and t = 2.0 can be used.

~TSFCS ~2 (aTSFC S yJ'aw Wf + aF F

dfTSFCf N N

eTSFC S ~4 (aTSFC S )4 (I-14)

aw Wf aF N FNf+ dfFd f wf

N

(I-IS)

(

1--x10,000

50\2

+ {-:.1o,ooo x 37.8\ 212

) \10,000 2 ') J(_1_ X 50\4 (-10,000 x 37.8\ 410,000 'J 10,000 2

')"'-:'--'--60-~-+ 57

= no

The t value is 2, and the uncertainty is

U = ±(B+t95S) = ±[O.OO53 + (2.0)(0.0063)] = ±0.01791bm/lbf-hr

The results of the error analysis are presented in Table IV.

The uncertainty limi~ as a percent of the nominal value may be calculated bydividing the uncertainty limit in engineering units by the corresponding nominal valueand then multiplying by 100.

The propagation of error formulas used in this section are derived and discussed inAppendix B.

12

AI: D C-TR-73-5

Table IV Uncertainty Components

Nominal Degrees ofParameter Value Bias Limit Precision Error Freedom I Uncertainty ..

"--.,

Tluust, FN 10,0001bf 18.11bf 37.81bf 57 93.71bfFuel Flow, Wf 10,000 lbm/hr 501bm/lu 50lbm/lu 60 1501bm/luThrust Specific

1.0 lbm/lbf-lu 0.00531bm/lbf-lu 0.0063Ibm/lbr-lu iJlO 0.018Ibm/lbf-luFuel Consumption

1.6 MEASUREMENT PROCESS

In making uncertainty analyses, definition of the measurement process is of utmostimportance. Uncertainty statements are based on a well-defined measurement process. Atypical process is the measurement of thrust specific fuel consumption (TSFC) for a givengas turbine engine at a given test facility. The uncertainty of this measurement processwill contain precision errors due to variations between installations, test stands, andmeasurement instruments. This uncertainty will be greater than the uncertainty forcomparative tests to measure TSFC on a single test stand for a single engine, a differentmeasurement process. The singl~stan~,single~ngine, back-to-backtes! \V()uld assullleJl1atIl1()st ..'inst~ll~.t.!()~ ....cm<isarrbration~rro~s \Vowd be ~ias~~. rat~~r ..t~~n. precision .errors.niases' mai be ignored in comparative' testing in that the same' equipment is used for alltesting, and biases do not affect the comparison of one test with another (the testobjective being to determine if a design change is beneficial)..!f1.e~ing1e stand, singleengine model and (>ther comparative tests are treated in S~ction 8.3.

---',,-. . "-., •.._._ •._-"..•~ ,-~-~ • ,

Because the definition of the measurement process is a prerequisite to defining themathematical model, all the elemental bias and precision error sources which affect themeasurements must be listed. Then, it must be determined how the bias and precisionerrors are related to the engine performance parameter. Based on this definedmeasurement process, the errors may be biases or precision errors.

The bias and precision errors related to the defined measurement process for thrustspecific fuel consumption are listed in Section II. Uncertainty analyses should be repeatedperiodically. Continuous validation is essential.

1.7 REPORTING ERROR

The definition of the components, bias limit, preCISIon index, and the limit (U)suggests a format for reporting measurement error. The format will describe thecomponents of error, which are necessary to estimate further propagation of the errors,and a single value (U) which is the largest error expected from the combined errors.Additional information, degrees of freedom for the estimate of S, is required to use theprecision index. These numbers provide all the information necessary to describe and usethe measurement error. The reporting format is:

1. S, the estimate of the precision index, calculated from data.

2. df, the degrees of freedom associated with the estimate of the precisionindex (S).

13

AE DC-T R-73-5

3. B, the upper limit of the bias error of the measurement process or B- andB+ if the bias limit is nonsymmetrical.

4. U = ±(B + t95 S), the uncertainty limit, beyond which measurement errorswould not reasonably fall. The t value is the 95th percentile of the two-tailedStudent "t" distribution.

5. U, the interval between U- = B- - t9 5Sand U+ = B+ + t9 5S. These limitsshould be reported when the bias limit is nonsymmetrical'

The mood components, S, df, B, ana U, are required to report the error of anymeasurement process. As recommended in Section 1.4, for simplification, the first threecomponents may be relegated to the detailed sections of uncertainty reports andpresentations. The first three components, S, df, and B, are necessary to propagate theerrors further, to propagate the uncertainty to more complex parameters, and tosubstantiate the uncertainty limit.

1.8 TRACEABILITY

In recent years the demanding requirements of military and commercial aircraft haveled to the establishment of extensive hierarchies of standards laboratories within themilitary and the aerospace industry. The NBS is at the apex of these hierarchies,providing the ultimate reference for each standards laboratory. It has becomecommonplace for Government contracting agencies to require contractors to establish andprove traceability of their measurements to the NBS. This requirement has created evenmore extensive hierarchies of standards within the individual standards laboratories. Ateach level of these hierarchies, formal calibration procedures are used. These proceduresnot only define calibration methods and intervals but also specify just what informationmust be recorded during a calibration, Le., meter model, serial number, calibration date,etc., in addition to actual measurement data.

The measurement process takes place over a long period of time. During this period,many calibrations occur at each level. Therefore, the precision errors of each comparisonare precision errors affecting the measurement process. The overall effect on themeasurement of force is a random (precision error) one. Therefore, the resultant overallprrcision index is the root-sum-square of the individual precision indices. For eachcomparison, the resultant calibration value is usually the average of several readings. Theassociated .precision index would be a standard error of the mean (or standard error ofestimate) for that number of readings. The precision index is

(1-16)

for four steps in the calibration process.

The degrees of freedom for each precIsIon index may be combined using theWelch-Satterthwaite formula to provide an estimate of the degrees of freedom for thecombined precision index.

14

AEDC-TR-73-5

(1-17)

This technique was simulated for various sample sizes and provides the best knownestimate of the equivalent degrees of freedom. The results of the simulation and thefurther use of the technique are discussed in CPIA No. 180 (AD855130). If non-integralvalues of df result from the Welch-Satterthwaite estimate, appropriate Student's t valuescan be found by interpolating from the table in Appendix E.

The unknown bias error limit for the end instrument is usually a function of manyelemental bias limits, perhaps ten or twenty. It is unreasonable to assume that all of thesebiases are cumulative. There must be a cancelling effect because some are positive andsome are negative. For this reason, the arbitrary rule that the bias limit B will be theroot-sum-square of the elemental bias limit estimates was adopted:

(1-18)

(~here L is the number of sources of bias.)

In combining elemental nonsymmetrical bias limits, the upper limits should beroot-sum-squared to determine the combined upper limit. The lower limits should beroot-sum-squared to determine the combined, lower limit. The resulting will benonsymmetrical bias limits. An example of an error analysis containing nonsymmetricalbias limits is given in Section VIII.

The uncertainty in the measurement instrument due to calibration is calculated usingthe uncertainty formula:

U = ±(B + t955)

where S is the precision index calculated from Eq. (1-16).

List of References on Statistical Quality Control Charts

Basic References

(1-19)

"ASTM. Manual on Quality Control of Materials." ASTM STP 15-C (Available fromAmerican Society for Testing Materials, 1916 Race Street, Philadelphia, Pennsylvania19103).

ASQC Standard Bl, B2-1958. "Guide for Quality Control and Control Chart Method forAnalyzing Data." ANSI Standard ZI.1, Z1.2, 1958.

15

AEDC-TR-73-5

ASQC Standard B3-l958. "Control Chart Method of Controlling Quality duringProduction." ANSI Standard Zl.3, 1958. (Available from American Society forQuality Control, 161 West Wisconsin Avenue, Milwaukee, Wisconsin 53203 or fromAmerican National Standards Institute, 1430 Broadway, New York, New York,10018).

Duncan, A. J. Quality Control and Industrial Statistics. Third Edition, Richard D. Irwin,Incorporated, Homewood, Illinois, 1965.

Cowden, D. J. Statistical Methods in Quality Control. Prentice-Hall, Incorporated,Englewood Cliffs, New Jersey, 1957.

Juran, J. M., Editor. Quality Control Handbook. Second Edition, McGraw-Hill BookCompany, New York, New York, 1962.

Examples of Control Charts in Metrology

Ku, H. H. "Statistical Concepts in Metrology." Chapter 2, Handbook of IndustrialMetrology. American Society of Tool and Manufacturing Engineers, Prentice-Hall,Incorporated, New York, New York, 1967. (Reprinted in NBS SP 300-Vol. 1,"Precision Measurement and Calibration-Statistical Concepts and and Procedures."Available from the Superintendent of Documents, United States GovernmentPrinting Office, Washington, D.C. 20402).

Pontius, P. E. "Measurement Philosophy of the Pilot Program for Mass Calibration."NBS Technical Note 288, (Available from the Superintendent of Documents, UnitedStates Government Printing Office, Washington, D.C. 20402).

Pontius', P. E. and Cameron, J. M. "Realistic Uncertainties and the Mass MeasurementProcess." NBS Monograph 103, (Available from the Superintendent of Documents,United States Government Printing Office, Washington, D.C. 20402).

16

AEDC-TR·73·5

SECTION IIUNCERTAINTY MODEL

2.1 GENERAL

Terms such as bias3 , precision error, uncertainty, standard deviation, NBS,traceability, calibration, and degrees of freedom and the statistical concepts andmathematical procedures to be employed were introduced in Section I. This section willdescribe the mathematical model with words, illustrations, and an example.

It is intended that the examples given will closely fit typical applications. However,the model is general, and if specific calibration hierarchies are more or less extensive thanthe examples, simply add or omit levels and apply the model as shown; iL~pecificmeasurement systems are different from the exa:q1p!es, substitute the" bias limit andprecIsIon indexletills'fot the. system c()mp9tleni~'a.nd apply the. model.

To review briefly, there are two types of measurement error: precision and bias.Precision error is the variation of repeated measurements of the same quantity. Thesample standard deviation (S) is used as an index of the precision. Bias is the differencebetween the true value and the average of many repeated measurements. A limit (B) forthe bias is estimated based on judgment, experience, and testing. The formula forcombining these into uncertainty (D) is

(II-I)

or

when nonsymmetrical biases are present.

~~N<:>te "that throughout this HandboOJ( lower case notation always indicates elementalerr9rs, i.e., sand b for elemental precision and bias, an.dllPper case notation indicates the

_____ ~.. ....-~ _.' _". •... •••• ..•.•. . ..' .' . . . •...• - ." .- • ...• .' •••• d" •• ' • '. ••.• ••.• . .' '" '.' ..

_~g<:>~~1.l111::~q1.f3:'i~·(:It~S)combination ofseveral errors,~~~., ..

where

±~Y"""i Si

±~Y"""i Di

3For a definition of terms used in this Handbook, see the Glossary in Section IX.

17

AE DC-TR -73-5

The remainder of this section is devoted to illustration of a typical measurementuncertainty analysis and the propagation of errors to performance parameters.

2.2 MEASUREMENT ERROR SOURCES

For purposes of illustration, the elemental error sources for the force measurementsystem will be treated in this section. These error sources fall into three categories:

1. Calibration Hierarchy Errors

2. Data Acquisition Errors

3. Data Reduction Errors

(2.2.1 )

(2.2.2)

(2.2.3)

Elemental error sources for other measurements will be enumerated in the sectiondealing with each measurement.

Calibration Hierarchy Errors

To demonstrate traceability of measurements to the NBS, whose standards are bydefinition the "truth," it is necessary to establish calibration hierarchies. Each level in thehierarchy, including NBS, constitutes an error source which contributes to the error inthe final measurement. Calibration of all measurement instruments at the NBS is possible;

National Bureau of Standards (NBS)

Inter-Laboratory Standard (ILS)

Transfer Standard (TS)

Working Standard

MeasurementIns trumen t (MI) ........~__--' ........__--' ........ ........11

Fig. 11-1 Force Measurement Calibration Hierarchy

however, such calibrations would be inconvenient, time consuming, and very expensive.The purpose here is to illustrate a typical hierarchy and to enumerate the error sources

Table V Calibration Hierarchy Error Sources

Bias Precision Degrees ofCalibration Limit Index Freedom

NBS - ILS bu Su dfuILS - TS b21 S21 df21TS-WS b31 S31 df31WS-MI b41 S41 df41

within. Figure II-1 is a typical' forcetransducer calibration hierarchy. Associatedwith each c<;>mparison in the cali~ration

hierarchy is a pair of elemental errors. Theseerrors are the unknown bias and theprecision index in each process. Note thatthese elemental errors are independent, e.g.,b2 1 is not a function of b11. The errorsources are listed in Table V.

18

AEDC-TR-73-5

2.2.2 Data Acquisition Errors

Data are acquired by measuring the electrical output resulting from force applied toa strain-gage-type force measurement instrument. Figure 11-2 illustrates some of the errorsources associated with data acquisition. Other error sources such as electrical simulation,thrust bed mechanics, and environmental effects are also present. The best method todetermine the effects of all of these error sources is to perform end-to-end calibrationsand compare known applied forces with measured values. However, it is not alwayspossible or even desirable to do this, and if this is the case, it is necessary to evaluateeach of the elemental errors and combine them to determine the overall error.

ForceTransducer Excitation-=- Voltage-=- Source

SignalConditioning

Measurement Signal

Fig. 11-2 Data Acquisition System

RecordingDevice

All the data acquisition error sources are listed in Table VI. Symbols for theelemental bias and precision errors and for the degrees of freedom are shown.

2.2.3 Data Reduction ErrorsTable VI Data Acquisition Error Sources

Computers operate on raw datato produce output in engineeringunits. The errors in this process stemfrom calibration curve fits (Fig. II-3)and computer resolution.

Symbols for the data reductionerror sources are listed in Table VII.These errors are often negligible ineach process.

AppliedForce

Measured Force

Fig. 11-3 Calibration Curve

Bias Precision Degrees ofError Source Limit Index Freedom

Excitation Voltage b12 S12 df12Electrical Simulation b22 S22 df22Signal Conditioning b32 S32 df32Recording Device b42 S42 df42Force Transducer bS2 SS2 dfS2Thrust Bed Mechanics b62 S62 df62Environmental Effects b72 s72 dh2

Table VII Data Reduction Error Sources


Calibration Curve Fit bI3 s13 df13Computer Resolution b23 S23 df23

19

AEDC-TR-73-5

2.3 MEASUREMENT UNCERTAINTY MODEL

The measurement process, defined for an entire engine test facility, is the totality ofall the individual subprocesses and steps in the measurement system for a given enginetype. That is, the process is the total of all the calibrations, all data acquisitions and alldata reductions. Therefore, the precision error in each step of each subprocess is reflectedas a precision error in the total process. The bias error in each subprocess is a bias errorin the total process. (Anoth~:r definition for the measurement process is discussed inSection VIII).

The precision index (S) at any stage in the total process is the root-sum-square ofthe elemental precision indices for that stage with the elemental precision indices for allof the preceding steps.

S (11-2)

where j defines the subprocesses calibration, data acquisition, and data recording and idefines the steps within the subprocess.

For example: The precision index for the calibration process is the root-sum-squareof the elemental precision indices of Table V.

S ..J 2 2, 2 2cal = ± 81l + 8 21 + 831 + 841 (11-3)

The precision index for the data acquisition process is the root-sum-square of theprecision indices of Table VI.

SDataAcquisition

± (11-4)

The precision index for the data reduction process is the root-sum-square of theprecision indices of Table VII.

(11-5)

The force measurement precision index is the root-sum-square of all the elementalprecision indices in the force measurement system.

The bias limit for any stage in the process is the root-sum-square of the elementalerrors in the preceding steps of the process.

20

AEDC-TR-73-5

For example: The bias limit for the calibration hierarchy is

The bias limit for the data acquisition process is

BData = ±~ b I22 + b22

2 + b322 + b42

2 + bS22 + b62

2 + b722

Acquisition

The bias limit for the data reduction process is

BData = ±~ bI32 + b23

2

Reduction

The bias limit for the force measurement process is

(II-7)

(11-8)

(II-9)

(II-II)

Biases associated with force measurement are equally likely in either the plus orminus directions, i.e., there are no nonsymmetrical bias limit estimates.

The degrees of freedom (df) associated with the precision index at any step in theprocess are calculated using the Welch-Satterthwaite formula. It is a function of the degreesof freedom and magnitude of each elemental precision index.

For example: The degrees of freedom for the calibration precision index (Sc al) is

r. 2 2 2 2]2~ll + s21 + s31 + s41 .

df == [s11 4 s 2 14 s 3 1 4 s 4 14J

dfll + df21

+ df31

+ df41

The degrees of freedom for the force measurement precision index is

(II-l 2)

The uncertainty parameter (D) at any stage is the sum of the bias limit (B) for that stageand the precision hmit (t95 S). The precision limit (t95 S) for any stage is the precisionindex (S) for that stage times the 95th percentile of the student's "t" distribution (whenthe degrees of freedom are greater than 30, 2.0 is used for the "t" value). Theuncertainty parameter (D) .defines the limits of the measurement error that mightreasonably be expected in a well-defined measurement process:

(11-13)

21

AEDC-TR-73-5

Figure II-4 illustrates the uncertainty parameter (V).

Measurement

~----Largest Negative Error-(B + t 95S)


. +(B + t95

S)

toe---- -B

Measurement Scale

Range of ±t95S--.............- Precision ---__....-

Error+B

1"'It-----------Uncertainty Interval---------........(The True Value Should Fall within This Interval)

Fig. 11-4 Uncertainty Parameter U = ±(B + t9SS)

2.4 EXAMPLE OF THE MODEL

Figure II-Sa is a block diagram showing the overall model for determining theuncertainty in gas turbine engine thrust specific fuel consumption. The blocks identifythe two major parameters: net thrust and fuel flow. The dotted lines indicate thecalculation of uncertainty for each parameter; solid lines indicate the propagation of biaslimits and precision indices to the bias limit and precision index for thrust specific fuelconsumption using Taylor's series methods. Detailed treatment of fuel flow measurementuncertainty and net thrust determination uncertainty is contained in Sections IV and VII,respectively.

In Fig. II-Sb, it is seen that each parameter block contains three general types oferrors: calibration errors, data acquisition errors, and data reduction errors. These areidentified by SI, S2, and S3, respectively, for precision indices and Bb B2, and B3,respectively, for bias limits. The lines within each parameter indicate the calculation ofuncertainty (VI, V2, and V3) for each type of measurement. Other lines indicate thecalculation of bias limits and precision indices for the individual parameters.

The final figure of the series (Fig. II-Sc) illustrates the fact that each measurement ismade up of several elemental sources of error. Examples of these are tabulated in TablesV, VI, and VII. In the figure, blocks indicate the formulas for the calculation of biaslimits and precision indices at each level in the measurement process. The lines point outthe procedures for determining bias limit (B), precision index (S), and uncertainty (V) foreach process in the measurement chain and also for the calculated parameter, thrustspecific fuel consumption.

22

AEDC-TR -73-5

IIL-------I

IIL--------t

II

L-----i

II

UWl r------- J

III

UTSFC 1--- - __ J

Bias Uncertainty PrecisionNET THRUST

~ ~(See Section VII)~ ~

a. General View b. Propagation of Errors

c. Elemental ErrorsFig. 11-5 Overall Uncertainty Model

23

AEDC-TR·73·5

As an example, the uncertainty in the thrust specific fuel consumption of a gasturbine engine will be calculated. Test conditions are that the engine is being tested on anoutdoor sea-level test stand. Assume a nominal sea-level thrust specific fuel consumptionof 1.01bm/lbf-hr and 10,000 Ibf and 10,000 Ibm/hr for net thrust and fuel flow,respectively. Detailed treatment of the errors in net thrust determination may be foundin Section 3.3 for sea-level testing which applies to this example. Section VII details thetreatment of errors in net thrust determination at altitude conditions. For simplicity,values for the bias and precision index for fuel flow have been assumed.

2.4.1 Net Thrust Measurement

I t is assumed that:

1. The net thrust bias limit is

BF ±18.11bfN

2. The net thrust precision index is

SF = ±37.8 IbfN

3. The net thrust nominal level = 10,000 Ibf

Then,

4. Net thrust uncertainty is

2.00 because df> 30 for SFN

U F ±(18.1 + 2.00 x 37.8)N

±93.7 Ibf

2.4.2 Fuel Flow Measurement

It is assumed that:

1. The fuel flow bias limit is

B W ±50 Ib/hrf

2. The fuel flow precision index is

Sw f = ± 50 Ib/hr, df w f 60

24

3. The fuel flow nominal level = 10,000 lb/hr

Then,

4. Fuel flow uncertainty is

U Wf = ±~W£+ t95SWf),t95

Uw = ±(50 + 2.00 x 50)£

±150 Ib/hr

2.4.3 Thrust Specific Fuel Consumption

The TSFC bias limit is

2.00 because df> 30 for S Wf

+ AI(1 x 50) 2 + (-10, 00 0 x 18.1) 2

- "\10,000 10,000 2

= ±0.0053 Ibm/lbf-hr

The TSFC precision index is

STSFC = ±( )

2-WI

+ -SFF 2 N

N

± + (-10 ,000 x 37.8\2

10,000 2 ))

±0.0063 Ibm/lbf-hr

The TSFC degree of freedom is

25

AEDC~TR~73-5

~~hen .the degrees o~ freedom f()r ~~cll source of err()r are gr~ater_~than~ ..thit~t¥T~th~t ___[~~tll tal1t~~~~es .._()(J!~Q.2mwiU.als()b~.. gr~a1~r.t~aI!thirJY ... :fi'()r .. illustrati"e .purposes, thecalculation of degrees of freedom are: " _.- ... ~.. - .--".~----~._._-~.------

~(1 50\2

+ (-10,000 x 37.8\ 21

2

~ 10,000 x) 10,0002 'j J(_1_ X 50)4 (-10,00~ x 37.8\ 4\10,000 -. 10,000 'j~~-60---<'- + 57

= 110

The result is greater than thirty, as expected. Therefore, t95 = 2.00.

The TSFC uncertainty is

U TSFC

U TSFC

2.5 SUMMARY

±(0.0053 -} 2.00 x 0.()063)

±0.018Ibm/lbf-hr

The statistical concepts and mathematical procedures used to develop the models areset forth in Section I.

In Section II, the uncertainty model is presented in mathematical, graphical, andblock diagram form with a numerical example of how the model is to be used. Thesemethods are summarized in Fig. 11-6, a logic decision diagram. However, Sections I and IIby no means provide full treatment of the problem of determining uncertainty in the gasturbine engine performance parameter, thrust specific fuel consumption. Some thingswhich have not been treated in this section are:

1. Multiple measurements (see Appendix III)

2. Evaluation of elemental errors (see Sections III through VII)

3. Signed bias (see Section I).

26

AEDC·TR·73·5

To Estimate Use Formula

Bias Limit (For Treatment of Signed Biases, see Section 1. 3.2)

1. Elemental (b) Judgment Supported by Estimate a Reasonable LimitSpecial Test Data for Each Bias Error

2. Measurement Estimated Elementals B . ...~Bias J i i

F DenotesMeasurement Bias and E(2!ar3. Performance the Propagation of B ... PerformanceParameter F j oj J ParameterBias Error (Taylor Series) Function

Precision Index l; (XL-X) 2L1. Elemental si Data from Multiple Si = NL-l , df ... NL-l

Measurements

2. Measurement Calculated Elementals S. =Ft[f s~r

df=~Precision and Data J i i'l; Si

Index i df i

J~DSj2

S ...3. Performance Measurement Precision •F

Parameter Indices and the Prop-Precision aga tion of Error

2] 2Index (Taylor Series)[ ~ (OF)S.

dfF

... ]. oj J

E[(~~ Sjr]J df.

J

t 95 Value Degrees of Freedom Interpolate in Two-TailedLess Than 30 (df < 30) Student's "t" Table for t

Degrees of Freedom Use t =2.0Greater Than or Equalto 3Q (df ~ 30) _.-

Uncertainty

1. Elemental Elemental Bias Limit Ui =±[Bi + t 95Si ]and Pre~ision Index

2. Measurement Measurement Bias U. =±[Bj + t95Sj]Limixs and Precision J

Indices

3. Performance Performance Parameter UF =±[ BF + t 95SF]Parameter Bias Limit and

Precision Index

Fig. 11-6 Logic Decision Diagram

27

AEDC-TR-73-5

SECTION IIIFORCE MEASUREMENT

3.1 GENERAL

The purpose of this section is to illustrate the measurement uncertainty4 model asapplied to a typical force measuring system. Figure III-I illustrates the system and will bethe basis for all examples in this section. The propagation of force measurement andother parameter measurement uncertainties to obtain net thrust uncertainty are discussedin Section VII.

Gas'furbine

Engine

Electrical CalibrationEquipment

Signal ConditioningEquipment

Force CalibrationSystem

Data ReductionEquipment (Computer)

Fig. 111-1 Force Measurement System

4Por a definition of terms used in this handbook, see the Glossary in Section IX.

29

AEDC-TR-73-5

3.2 FORCE MEASUREMENT ERROR SOURCES

Error sources for any measurement fall into three categories: (l) calibration, (2)data acquisition, and (3) data reduction. These categories will be discussed in this orderdescribing methods for determining values for the elemental errors.

3.2.1 Force Transducer Calibration Hierarchy

Figure 111-2 illustrates a typical force transducer calibration hierarchy.

National Bureau of Standards (NBS)

Interlaboratory Standard (ILS)

Transfer Standard (TS)

Working Standard

MeasurementInstrument (MI)

Fig. 111-2 Force Transducer Calibration Hierarchy

Bias Precision Degrees ofCalibration Limit Index Freedom

NBS -ILS bll Sll dfllILS - TS b21 S21 df21TS-WS b3l S31 df3lWS-Mr b41 S41 df41

The bias limit for the calibrationprocess is 'the root-sum-square of the elemental errors in the preceding steps of theprocess.

Associated with each comparison in the calibration hierarchy is a pair of elementalerrors.. These errors are the unknown bias and the precision index in each process. Notethat these elemental errors are not cumu-lative, e.g., b21 is not a function of b1l. Table VIII Calibration Hierarchy Error Sources

The error sources are listed in Table VIII.

The precision index for .the calibration process is the root-sum-square of theelemental precision indices in the preceding steps of the process.

(111-2)

Calibrations are accomplished by applying known forces to the instrument beingcalibrated and recording the output. The output may be inches deflection, millivolts,

30

AE D C-T R-73-5

pounds force, or other measurement units. Several known forces (usually eleven or more)are applied over the range of the instrument being calibrated as indicated in Table IX.The forces are applied first going up scale and then going down to determine thehysteresis errors.

Table IX Calibration Data

Calibration One Calibration Two

Applied Force Measured Force Calculated Force Measured Force Calculated ForceLevels, F (From Curve Fit (From Curve Fit

Up Down of Call Data) Up Down of Cal 2 Data)

0 0.00 0.20 0.105 0.00 0.20 0.112 1.00 1.60 1.30 1.25 1.50 1.384 2.20 3.10 2.65 2.50 2.90 2.706 3.75 4.82 4.29 4.00 4.60 4.308 5.77 6.77 6.29 5.90 6.50 6.20

10 9.30 9.30 9.30 9.29

The data from Table IX are plotted in Fig. 111-3. The calculated values aredetermined from a polynomial curve fit of the data. All four curves demonstrate thenonlinearity of the calibration (exaggerated for this example). The difference between theupscale lines and the downscale lines demonstrates the hysteresis of the system. Note thatthe contribution to overall uncertainty by transducer nonlinearity and hysteresis can beminimized by considering system performance over a relatively small range near nominal.

10

9

8

Po4 7

Q)

C,) 6~

o Calibration No. 1P<-4 5'0Q)

~:;:j 4UlellQ)

:::E: 3

2

1

1 2 3 4 5 6 7 8 9 10Applied Force, F

Fig. 111-3 Calibration Curves

31

AE DC-t R-73-5

3.2.1.1 Precision Index

A least squares polynomial curve (Y) is fitted to the calibration data:

Y = Ao + A 1F + A 2F2 + .•• + AkFk (111-3)

where Y = calculated force with F pounds force applied

k =degree of curve fit, Le., largest exponent of F

Ao through Ak are curve coefficients.

(1114)

N~ - 2~ (Y. - Y)

Jj=l

N-1

y

Figure 111-4 illustrates a plot of calibration data with the dashed line representing aleast squares polynomial curve fit. The Yvalue is the force calculated from the curve fitand corresponding to the applied force F inEq. (111-3). The Yj values are calculated fromcurve fits of previous calibrations establishinga set of Yj, j = I, ... N, where each Yjrepresents a calibration and curve fit. Theprecision index at any particular point Fi onthe curve is calculated by

Applied Force, F i

fig. 111-4 Scatter in Measured force

where Y = the average of all Yj values for this transducer at Fi

Si = the precision index for Y with Fi pounds force applied

N =the number of calibrations in the estimate

Equation (111-4) yields the precision index for any value of Fi for any number ofcalibrations. For example, Fig. 111-3 exhibits the data from two· calibrations of onedevice, sayan interlab standard. Table IX lists calculated values of Y from the data fromboth calibrations. At Fi = 4, Y = 2.675 and the Yj values are 2.65' and 2.70. These dataalong with Eq. (1114) yield

S.1

±(2.65 - 2.675)2 + (2.70 - 2.675)2

(2-1)

Note that Eq. (1114) yields the precision index for each value of Fi, but there are manyvalues of Fi. The precision index for a given Fi value will apply over a narrow range of F,possibly ±I0 percent of full scale from the point of interest. Generally, it should not beassumed that the precision index for 80 percent of full scale is the same as that for 10,20, 30 percent and so on. Equation (111-4) applies in all of these cases, but the

32

AE DC-T R-73-5

differences (Yj - y)2 must be summed for the different values of Fi and the largestreported as the precision index for the calibration process. Equations (111-3) and (111-4)and all of the rules presented above will apply to each level in the calibration hierarchy.The precision index for the complete hierarchy is calculated as indicated by Eq. (111-5):

where S1 = calibration hierarchy precision index

Si = precision index for the individual levelsin the hierarchy

(111-5)

3.2.1.2 Degrees of Freedom

The degrees of freedom (dfi ) associated with the precision index (Si) is one less thanthe number (N) of Y observations used to determine the precision index:

dfi = N-l (111-6)

Degrees of freedom (dfi ) may be calculated at each level in the calibration hierarchy.When the precision index (S1) is calculated for the complete hierarchy, also calculate df1for the hierarchy utilizing the Welch-Satterthwaite technique (see example in Section 2.3)if any of the degrees of freedom are less than 30.

3.2.1.3 Bias

The bias for each level in the calibration hierarchy should be reported as plus orminus the largest unknown fixed error expected. At the frrst level in the hierarchy (NBSversus interlab standard), the NBS will state an upper limit of bias for the deadweights,

LargestNegative

Bias

-b.1.

LargestPositive

Bias

+b.1.

Distributionof CalibrationData

Measured Force ~

Fig. 111-5 Calibration Hierarchy Elemental Bias

33

AEDC·TR·73·5

e.g., "the errors of the applied loads did not exceed 0.002 percent," quoted for thecalibration of an interlab standard with a 60,000-lb capacity. This, however, is only thebias in the applied forces. There will be additional biases resulting from the calibrationprocess. The estimate of the bias limits for the calibration process must be based oncareful analysis of the calibration data and any other available information tempered byengineering judgment. For example, the data from an extremely large calibration historymay lead to a bias estimate no greater than that reported at the preceding level in thehierarchy. On the other hand, if only one calibration is available to use as a guide to thebias estimate, the estimate from the preceding level in the hierarchy might be increasedby an order of magnitude.

When bias limits (bi) have been established for each level of the calibrationhierarchy, a bias limit Bl for the total hierarchy may be calculated, Le.,

(III-7)

3.2.1.4 Uncertainty

Uncertainty in the calibration process is now obtained by a simple combination ofthe precision index and bias limit (see example in Section 2.3).

Measurement

10+--- Largest Negative Error-(Bl + t9SSl)

Largest Positive Error---.~

+(Bl + t9SSl)

Measurement Scale

Range of ±t9SSl......- ....... -B.l-.........,~- Precision---..........-- +Bl ---.111i111

Error

......------Uncertainty Interval----------lllllot(The True Value Should Fall w1th1n This Interval)

Fig. 111-6 Calibration Process Uncertainty Parameter U1 = ±(B1 + tg5 S1)

As indicated in Fig. 111-6,

(111-8)

34

AEDC~TR"73·5

where U1 = calibration hierarchy uncertainty

B1 = calibration hierarchy bias limit

S1 = calibration hierarchy precision index

t95 = 95th percentile of the "Student's" t distributionwith df1 degrees of freedom

3.2.2 Data Acquisition and Reduction Errors


Excitation Voltage b12 S12 df12Electrical Simulation b22 S22 df22Signal Conditioning b32 S32 df32Recording Device b42 S42 df42Force Transducer b52 S52 dfs2Thrust Bed Mechanics b62 S62 df62Environmental Effects b72 S72 df72

The best method of evaluating the net effect of data acquisition and reductionerrors is to periodically perform applied load tests. These tests will evaluate all dataacquisition and reduction errors in-cluding errors due to force transduc- Table X Data Acquisition Error Sourceser temperature variations, fuel linetemperature and pressure variations,fuel flow and environmental effectson the thrust bed.

All data acquisition errorsources are listed in Table X. Symbols for the elemental bias andprecision errors and for the degreesof freedom are also shown.

The bias limit for the dataacquisition process is

(111-9)

The precision index for the data acquisition process is

(III-I 0)

The computer operates on the raw data to produce output in engineering units. Theerrors in this process stem from the calibration curve fits and the computer resolution.

Symbols for the data reductionerror sources are listed in Table XI.These errors are often negligible ineach process.

The bias limit for the datareduction process is

Table XI Data Reduction Error Sources


Calibration Curve Fit b13 s13 df13Computer Resolution b23 ~23 df23

\"

35

AEDC·TR·73·5

The precision index for the data reduction process is

(111-12)

3.2.2.1 Appl ied Load Tests

Applied load tests are performed as follows:

1. Install an engine in the test stand.

2. Connect all service lines (fuel, instrumentation, water, etc.) and otherrestraints to the engine.

3. Perform an end-to-end calibration of the force measurement system in theusual manner (with all service systems operating at nominal levels ifpractical).

4. Evaluate the calibration data and perform a curve fit according to normalprocedures.

5. Perform the usual pre-run set-up procedures as if preparing for an enginerun. Just prior to an engine run is an excellent time to perform an appliedload test.

6. Apply a known force by means of the force calibration system equal tothe nominal expected when the engine is delivering rated thrust.

7. Record digital. data at a sampling rate and for a period which is normal forsteady-state engine conditions; also, record the applied force (X) indicatedby the standard.

8. Make several (ten or more) recordings as defined in (7) above.

9. Reduce the data by means of the engine data reduction program.

10. Regardless of the number of measuring devices used (multiple bridge loadcells, multiple load cells, etc.), calculate multiple sample average Ykj .foreach steady-state recording. The average of the ten or more recordings (Yj)for the jth bridge or transducer is

Y.]

M

~Yk'k~l J

M

(111-13)

where M = number of steady-state recordings.

36

AEDC-TR-73-5

The grand average Y is then calculated for all load cells

Nl Y.i=l J

N

where N = number of bridges

(111-14)

11. This grand average (Y) represents a measurement of the known input. Theprecision index (Si) for this average is estimated from the precision errorof the bridges and/or load cells (Sb):

(111-15)

The estimate of Si is Sb divided by the square root of the number ofbridges (N):

s· = ±1 {N

± (111-16)

12. If several applied load tests are performed, the precision index of the dataacquisition and reduction process is calculated by pooling the estimates(Si) from each test:

(111-17)

where K tests have been performed.

13. The bias limit for data acquisition and reduction may b,e estimatedthrough careful analysis of ancillary data such as applied load test historiestempered by the judgment of the most knowledgeable force measurementengineer.

Applied load tests performed in the preceding manner will evaluate the net effect ofthe following error sources:

1. Excitation voltage

2. Electrical calibration of the data recording system

3. Analog to digital conversion

37

AE D C-TR -73-5

4. Stand mechanics

5. Calibration curve fit

6. Computer resolution

Performance of additional applied load tests at the conclusion of engine runs willprovide evaluation of the net effects of:

1. Environmental effects on force transducer

2. Effects of fuel line temperature and pressure variations

3. Environmental effects on the thrust bed

Post-run applied load tests are performed immediately after engine shutdown asfollows:

1. Ensure that fuel lines are at nominal run temperatures and pressures

2. If practical, flow fuel at nominal flow rates

3. Ascertain that the force measurement system is at run temperature

4. Apply a known force as described in the pre-run applied load test

5. Record test data as described in pre-run test procedures

6. Record an electrical calibration of the data recording system to ensure nochange since the pre-run calibration

7. Reduce data as described in pre-run test procedures

8. Calculate precision index using Eqs. (III-IS) through (III-17)

9. Estimate bias limit

3.2.2.2 Elemental Error Evaluation

If it is undesirable or impractical to perform applied load tests frequently, thealternative is to evaluate individually all elemental errors and combine them statistically.The complete list of data acquisition and reduction elemental errors is:

1. Stand mechanics

2. Fuel line temperature .variations

38

AEDC-TR-73-5

3. Fuel line pressure variations

4. Force transducer temperature variations

5. Excitation voltage

6. Recording system electrical calibration

7. Analog-to-digital converter nonlinearity and drift

8. Recording system resolution

9. Electrical noise

10. Tare evaluation (thrust stand losses)


In the following sections, the above errors are discussed in enough detail to evaluate themand combine them statistically.

3.2.2.2.1 Stand Mechanics

Several mechanical features of the thrust stand (Fig. III-7) must be considered andtheir effects evaluated. These features are:

1. Design of thrust bed support system

2. Flexure design

3. How much deflection can be expected in the thrust bed and measurementlinkage when a force equal to nominal rated thrust is applied at thepoint(s) of engine support?

Thrust Stand

,Engine ThrustI Restraint

Calibration Load Cell (2) L Data(Working Standards) Load

Pull Rod (2) Cell

Fig. 111-7 Gas Turbine Thrust Measurement System Calibration Configuration

39

AEDC-TR-73-5

4. How are fuel lines supported?

5. How are fuel lines and other restraints oriented with relation to the axialforce vector?

6. What is the effect of engine and instrumentation weight on forcemeasurement?

Some of these are simply considerations to be made, with improvements to minimize theeffects being the obvious course of action. Errors caused by stand mechanics will alwaysexist to some degree.

If in-place calibrations are not performed, i.e., only laboratory calibrations of theforce measurement transducers are performed, then applied load tests are required toevaluate errors caused by stand mechanics. For example, if applied load tests areperformed with and without the weight of the engine and instrumentation on the thrustbed and the load cell output is measured by means of a laboratory potentiometer toeliminate recording system errors, then the difference between measured force andapplied force with no engine installed is the error due to thrust bed design. Thedifference between measured force and applied force with an engine and instrumentationinstalled is the error caused by test stand design and the additional weight. If several testsof this nature are performed, distributions of data can be developed and handledstatistically. Figure 111-8 illustrates data distribution Xl without the engine installed, and

AppliedForce

WithoutEngineInstalled~-----..J

Measured Force ---....,1\111\11-

Fig. 111-8 Precision Errors

data distribution X2 with the engine installed. The precision index for tests without theengine installed is

'.

- 2~(Xi - Xl)

(N-I)

40

(111-18)

AEDC-TR-73-5

The precision index for tests with the engine installed is

where: x =applied force

Sl = precision index of tests without engine installed

S2 = precision index of tests with engine installed

Xl = the average of all measured values withoutengine installed

X2 = the average of all measured value withengine installed

N = number of tests with engine installed orwithout engine installed

(111-19)

Unknown bias limit estimates may be made as indicated in Section 3.2.2.1, item 13.Similar tests can be designed to evaluate the effect of other error sources in this category.

3.2.2.2.2 Fuel Line Temperature Variations

Variations in fuel line temperature result in more or less restraint on the forcemeasuring system, side loads, and possible axial loads. In-place calibration of themeasurement load cell will not account for errors from this source, unless, of course, runtemperature is maintained during calibration. Again, applied load tests are the bestsolution with data being evaluated as in the preceding section on stand mechanics. Ifin-place force transducer calibrations are performed, with the engine and all associatedplumbing installed, errors from the above sources will be reflected as nonlinearity andhysteresis in the calibration data. These errors then become a part of the error attributedto the working standard versus force measurement transducer calibration process.Calculation of the precision index (Si) at each operating point F is accomplished with Eq.(111-4). .

s·1

N _ 2I (Y. - Y)j=l J

N-I(111-4)

where Y is the value of the calibration curve at the operation point F, and the N values(Yj) are the calculated values at F as defined in Section 3.2.1.1 (Fig. 111-4). The degreesof freedom associated with Si are N-l.

41

3.2.2.2.3 Fuel Line pressure Variations

Side loads and axial loads are the usual result of pressurizing fuel lines. Once again, in-place calibration will not account for this error unless the lines are pressurized during the calibration. Applied load tests will aid in evaluating this error. Data should be evaluated as discussed in Section 3.2.2.2.1.

3.2.2.2.4 Force Transducer Temperature and Ambient Pressure Variations

3.2.2.2.4.1 Temperature Variations

The best method for minimizing the effect of temperature variations is to minimize the variations through environmental control. This, however, is not always possible. Another way to combat this problem is to perform applied load tests in the laboratory at different fixed temperatures on individual load cells. Suppose a load, cell is loaded to X pounds, N1 times, at a constant temperature of 75'F, and the average of the N observations is XI. The precision index and the bias, which is correctable, may be established as follows:

where S L ~ = precision index for the load cell

Xi = force indicated by the load cell

Now, if the same load X is applied, N2 times, at a constant temperature of 40°F and the average of the N2 observations is X2, then

where , S L ~ = precision index. for the load cell

Xi = force indicated by the load cell , *

To be conservative, the largest of the two errors (sL1 and is selected as the elemental error for this source. This will minimize the possibility that changes in conditions will invalidate the estimated uncertainty value. A relatively large change in bias and very little change in precision error is expected (Fig. 111-9).

A E D C-T R -73-5

In evaluating the effect of temperature variations on load cells, it is very important to deternine the temperature of the strain-gage bridge both in the laboratory and in the test environment. Large temperature gradients between the outer shell and inner working parts of a load cell may well result in serious errors in correction values.

A p p l i e d F o r c e

D i s t r i b u t i o n a D i s t r i b u t i o n a t Temwerature T, Tempera ture T2

I n d i c a t e d F o r c e - Fig. 111-9 f emperature Bias Effect on Distributions of Errors

3.2.2.2.4.2 Ambient Pressure Variations

When used in altitude test chambers, load cells may be exposed to a range of ambient pressure levels from approximately 14.7 (one atmosphere) to about 0.5 psia. A typical variation of load cell output versus ambient pressure is shown in Fig. 111-10. The sensitivity of the load cell to 4X

ambient pressure is, of course, a func- 2 Ambient Temperature

tion of the load cell physical characteris- 2: 3X Constant

tics. Some load cells are compensated to 5 '2 nullify partially the effects of ambient zx pressure variation. Typically, load cells 6 exhibit an increasing output in the x tension direction with decreasing ambi- a

ent pressure. Data reduction programs o must correct for load cell ambient 1/ 4 1/ 2 1

Atmosphere Atrnospher e Atmosphere

pressure effects to remove this bias error Ambient P r e s s u r e

from the data. The data reduction program typically makes this correction Fig. 111-10 Ambient Pressure Effect on based on a measurement of the load cell Load Cell Output environmental pressure during altitude testing.

To determine the load cell sensitivity to ambient pressure, laboratory tests are performed with. the load cell in a "beB1 jar." The pressure within the "bell Jar" is adjusted over the desired range, and load cell output is recorded at several discrete pressure levels. Each discrete pressure level is maintained .for several minutes, and the load cell output is monitored to detect any significant pressure leaks in' the load cell.

AEDC~TR-73-5

The elemental errors associated with the load cell ambient pressure calibration, theenvironmental pressure parameter measurement, and data reduction must be considered indetermining the force measurement uncertainty.

3.2.2.2.5 Excitation Voltage Errors

Instrumentation excitation power supplies are usually purchased with rigidrequirements on output voltage drift and regulation. If verification limits for excitationvoltage are established, say ±O.05 percent of the output, and then laboratory tests areperformed to ensure that the power supply produces the desired voltage within theselimits; then the precision index is

and the bias limit is

si = ±0.0003 x desired voltage

hi = ±0.0005 x desired voltage

(111-24)

(111-25)

where Si is the calculated precision assuming a uniform distribution over the interval±O.OOOS. Uniform is a conservative assumption for the errors. They probably will have acentral tendency.

For any uniform interval, the precision index is calculated by taking the square rootof the upper limit minus the lower limit squared divided by 12:

(See Appendix A for the derivation.)

(Upper - Lower)2

12(111-26)

Better estimates of the elemental bias and preCISlon errors can be obtained bymeasuring the excitation voltage during a run. This can be done by randomly selecting achannel to be reserved to measure the voltage during the run. The uncertainty limitdepends on the frequency of error variation in terms of test duration. The error could belargely precision (Fig. 111-11 a) and may be long term (drift) and/or short term (noise).

The error could be mostly bias (Fig. 111-11 b) with very little precision error (thiscould be long term drift which appears as bias during anyone test). Another possibility· isthat the system could contain both bias and precision error (Fig. III-lIe). The estimatesfrom the recorded data would be

N --~-=-2

~(x.- X)i=l 1

N·::r(111-27)

and hi .? ±Idesired voltage - XI (111-28)

where X = the average value recorded during the run.

44

AEDC-TR-73-5

Long Term (Drift)

Term (Noise)

\ I\ I\J

-0.005, ..t---------Test Dura tion-------~

a. Precision Error

Long Term (Drift)

Short Term(Noise) ,.,....

.......--------Test Dura tion -------~

-0.005

b. Bias Error

,*",I \I \

I \ I\ I '\,\ I 'v,)

" ~ Short" Term (Noise),\1\"I I \ i LOng Term__~~\ I \ (Drift)

I I \I \ \I \ I \

\ I \ I- V

o li""II....-------Test Duration ~

+0.005

c. Both Bias and Precision ErrorsFig. 111-11 Errors

45

AEDC·TR·73·5

3.2.2.2.6 Recording System Electrical Calibration

Recording system electrical calibrations for force measurements are performed byapplYing known voltages such as zero and full scale at the recording system input todetermine system sensitivity to load cell analog voltages. Another method of electricalcalibration is the shunt resistor calibration. This method employs precision resistorswhich, by means of switches, are placed in parallel with one or more of the active bridgelegs in the load cell. These shunt resistors unbalance the load cell bridge and· cause aresultant voltage output.

Errors associated with voltage calibration are:

Precision Index Bias

1. Calibration power supply

2. Analog to digital converSiOn}

3. System resolution

sp

Calibration power supply errors sp and bp are evaluated by the same procedure employedfor excitation power supplies (see Section 3.2.2.2.5).

The net effect of errors from the other two sources can be evaluated by measuringthe voltage output of the calibration power supply with a laboratory standard and therecording system. If this test is performed a number of times, a distribution can bedeveloped similar to the one shown in Fig. 111-12.

The precision index (Sk) is

where Xj = voltage indicated by the recording system

X = average of all recording system indications

N = number of observations of Xj

X = voltage indicated by the laboratory standard

(111-29) .

The electrical calibration precision index (Si) is

(111-30)

Estimating limits for the unknown biases bp and bk are left to the judgment of the mostknowledgeable engineer. Of course, this judgment should be supported by an extensivehistory of special tests designed for this specific purpose.

46

In the case of shunt resistorcalibration, recording system sensitivityis determined in terms of pounds perdigital count. This is accomplished byrecording the count level at somereference force level (for example, zeroapplied force) and then recording thecount level with the resistor shuntedacross the bridge. The force equivalentof the bridge unbalance caused by theshunted resistor is known. Therefore,the change in recording system countscorresponding to this force changedetermines system sensitivity (poundstechnique are

AE DC~TR·73-5

x = Average Measured Voltage

Measured Voltage •

Fig. 111-12 Elemental Precision Error ofCal ibration Power Supply

per count). The errors associated with thisI

Precision Index Bias

1. Erroneous Shunt Resistance Values

2. Line Resistance

3. Reference Errors sp

Analog-to.digital conversion and system resolution errors may be evaluated by thetechnique described below. The reference errors (sp and bp) are evaluated by pre- andpost-run tare investigations. The total effect of the first two error sources may beevaluated as follows:

1.- Install a force transducer and connect to the recording system

2. Install a force standard in the system such that it senses the same force asthe measurement transducer

3. Switch in the shunt resistor and make a multiple scan (ten or more)recordings (Yi)

4. Apply a force equivalent to the shunt resistor as indicated by the forcestandard and established by calibration

5. Make a multiple scan (ten or more) recording (Yk )

6. Repeat steps 3, 4, and 5 several times, say ten

7. Calculate a multiple scan average Yi and Yk for steps 3 and 5, respectively

8. Further calculate a Yi and Yk for the repeated recordings

47

AEDC·TR·73-5

The difference "Vi - "Vk is correctable. Estimates of the limits of the unknown bias fromthese sources are left to engineering judgment supported by special test data.

Applied Voltage X

Fig. 111-13 Sensitivity Curve

3.2.2.2.7 Analog-To-Digital Conversion Errors

This portion of the data acquisition systemincludes amplifier and analog-to-digital (A/D)converter. Consider these two components as a singleunit, first because modern data recording systemsemploy only one amplifier which is a part of the A/Dconverter, and secondly, because the errors are moreeasily evaluated.

Analog-to-digital conversion error is evaluatedby the following procedure:

1. Select a stable voltage source and a voltage standard

2. Connect the voltage source to the A/D converter such that it is commonto several channels

3. Connect the voltage standard into the system such that it will have noeffect on the recorded data

4. Make a multiple scan (ten or more) recording with the input to allchannels shorted, Le., with the input voltage equal to zero

S. Make a multiple scan (ten or more) recordings with the input at a levelwhich will produce a digital indication slightly less than full scale

6. The data from 4 and 5 above will allow calculation of the sensitivity (Fig.111-13) of each channel to applied voltages. For example:

S ... (counts) digital output (counts)ensltlvlty -_. = -.!.• ...;;;,....-.,..-.--=-:----:---volt inpu t (volts) (111-31 )

7. Apply discrete voltages (typically eleven or more) in ascending anddescending steps to the AID conversion unit input and make multiple scan(typically ten or more) recordings at each level

8. Repeat 7 several times over a period of several hours (typically four ormore). Calculate a multiple scan ~erage (Yij counts) for each channel ateach voltage level. The average (Xj) for each voltage level and for eachchannel is

y.]

M~ Y ..i=l 1J-_..

M

48

(111-32)

AEDC-TR-73-5

where M = number of multiple scan averages at each voltage level for eachchannel. The grand average (Y) at each voltage level is then calculated for allchannels

N:£ Y.i=l J

N

(111-33)

where N = number of channels. The precision index for AID conversion is thewithin channel variation pooled for the N channels:

±

N M.J _ 2

:£ :£(Y .. -Y.)j= 1 i= 1 1J J

N:£ (M. - 1)

j= 1 J

(111-34)

For an actual engine measurement, this preCISIon index must be converted to thestandard error of the mean by dividing by the square root of the number of recordingsused:

(111-35)

where K recordings are used for each measurement. This test will yield a conservativeestimate of the precision index because the precision index in the voltage standard willinfluence the data.

3.2.2.2.8 Recording System Resolution

This is primarily a fixed error in that for digital systems the resolution error is plusor minus one-half of one count. Whether the system is three digit or four digit, the erroris still one-half of one count. What really happens is that, if a system is between digits,e.g., 5000.5, the indicator lights will alternate between 5000 and 5001. If ten recordingsare made, the average will likely be 5000.5. However, suppose the system indicates aconstant digital value of 5001 for an analog input equal to 5000.6. Nothing has beengained from multiple sampling and averaging in this case. The precision index forrecording system resolution is

Si ±0.3 uni ts

and the bias limit is zero.

3.2.2.2.9 Electrical Noise

Electrical noise is an error which is generally purely random. The effect can best bedescribed as a variable indication of an input which is constant. The effect of electricalnoise can be minimized by making several measurements and averaging. Suppose the

49

AEDC-TR-73-5

digital indication of a constant 5-volt signal varies randomly over the interval from 4090to 5010. If ten measurements are recorded and averaged, the answer will be very close to5000. The precision error caused by electrical noise is

~N 2:£ (Y .... y)i=l 1

s· = ±1 N-l

(111-36)

where Si = electrical noise precision index

Yi =individual recorded values

Y =the average of all measurements

N = the number of measurements

3.2.2.2.10 Tare Variations

(111-37)

N-lTare---....1lP

If the measurement system output is recorded for each run .after engine shutdown,with fuel line temperatures and pressures at run conditions, with the force transducer and

thrust bed at run temperature and with fuelx (Average Tare) flowing at nominal rates if possible, most of

the bias caused by these conditions will beeliminated. Tare variation history on anygiven test stand should' Yield a datadistribution similar to Fig. 111-14. The

(Tare Variation precision index for tare is thenElementalPrecision Index)

Fig. 111-14 Tare History Showing Elemental Precision Error

The bias limits for tare measurement areestimated in the same manner as the biaslimit for force measurement.

3.2.2.2.11 Computer Resolution

Computer resolution is the source of a small elemental error. Some of the smallestcomputers used in experimental test applications have six digits resolution. The resolutionerror is then plus or minus one in 106 . Even though this error is probably negligible,some consideration should be given to it. For example, consideration should be given torounding-off and truncating errors. Examples of rounding-off are

10.4 is rounded off to 10.0

10.6 is rounded off to 11.0

SO

AE DC-TR -73-5

Rounding-off results in a precision error. Examples of truncating are

10.1 = 10.0

10.9 = 10.0

Truncating always results in a bias. Whenever faced with the choice of a small precisionerror versus a small bias, the precision error is generally chosen. Therefore, the computerresolution error is

si = ±O.3 digits (see Appendix A for derivation)

3.3 FORCE MEASUREMENT ERROR ANALYSIS

By assuming completely hypothetical numbers for the elemental error terms for thecalibration hierarchy, data acquisition, and data reduction processes, Table XII tabulatesvalues for all elemental bias and precision error terms as defined in Tables VIII, X, and"XI. Table XII also includes sample sizes for the calibration processes.

Table XII Force Measurement Elemental Error Values

Calibration Errors, lb Data Acquisition Errors, lb Data Reduction Errors, lb

Bias Precision Sample Bias Precision Bias PrecisionSize

bll = :to.2 Su = ±10.0 6 bl2 = ±S.O sl2 = ±S.O b13 = ±10.0 sl3 = negligibleb2l = :to.2 S2l = ±10.0 11 b22 = ±S.O S22 = ±S.O b23 = negligible S23 = negligibleb31 = :tOA s3l = ±14.1 5 b32 = ±S.O S32 =±S.Ob41 = :to.8 s4l = ±20.0 17 b42 = ±S.O S42 =±S.O

bS2 =:toA S52 = ±20.0b62 = ±10.0 S62 =±10.0b72 = ±S.O S72 =±S.O(df = 31 for all elemental

precision errors)

The errors associated with the calibration hierarchy, data acquisition, and datareduction stages in the measurement process are calculated below and are identified bySl, S2, and S3, respectively, for precision indices and Bl' B2, and B3, respectively, forbias limits and Ul, U2, and U3, respectively, for uncertainty intervals.

1. Calibration bias limit for the force transducer is

(III-38)

B 1 ±O.94 Ib

51

AI: DC-TR-73-5

2. Calibration precision index estimate for the force transducer is

(111-39)

SI ±28.3 1b

3. To demonstrate use of the Welch-Satterthwaite method fordetermining degrees of freedom (df), small sample sizes for thecalibration processes in the force transducer calibration hierarchyhave been assumed. Sample sizes are included along with theelemental errors in Table XII. From Section I,

df( 8 2 + s 2 + .. + s 2)2

1 2 n(111-40)

where dfn = sample size minus one for the nth calibration2

[(10)2 + (10)2 + (14.1)2 + (20)2]df1 =

(10)4 (10)4 (14.1)4 (20)4-5-+10.+ 4 +]:"6

640 X 10 3

23 X 10 3

27.8

Under the "t" column in Table E-l in Appendix E, t is 2.052 for 27degrees of freedom and' 2.048 for 28 degrees of freedom.Interpolating linearly gives a t of 2.049 for 27.8 degrees of freedom.

The calculation of calibration uncertainty (D1) for the forcetransducer is then

U 1 ±(B1 + t95S1)

±(0.94 + 2.049 x 28.3)

±58.9 1b

52

(111-41 )

AEDC-TR-73-5

4. Data acquisition bias limit is

±~b.2• II

±~ (5) 2 + (5) 2 + (5) 2 + (5) 2 + (0.4) 2 + (10) 2 + (5) 2

± 15.0 Ib

5. Data acquisition precision index estimate is

±~S.2• II

(11142)

.----------------- (11143)±..J (5)2 + (5)2 + (5)2 + (5)2 + (20)2 + (0)2 + (5)2

± 25.0 Ib

6. Data acquisition uncertainty is

±(8 2 + t9s52)

±05.0 + 2 x 25.0)

65.0 Ibt 9s = 2.00 because df> 30 for 52

7. Data reduction bias limit is

±~ rbi2

± 10.0 Ib

8. Data reduction precision index estimate is

±~S.2• II

o

9. Data reduction uncertainty is

U 3 ±(83 + t 9s53)

±(10 + 0.0)

± 10.0 Ib2.00 because df> 30 for 53

10. Force measurement bias limit is

±18.1 Ib

53

(11144)

(11145)

(11146)

(11147)

(11148)

11. Force measurement precision index estimate is

SF = ±~ S12+ S2

2+ S32

±~ (28.3)2 + (25)2 + 0 2

±37.8 Ib

12. Degrees of freedom for force measurement are

2(S 2 + S 2 + S 2)123

S4+ S 4+ S 4123---

df1 df2 df3

2[(28.3)2 + (25)2]

(28.B)4 + (25)427.8 31

57

(111-49)

(III-50)

13. Force measurement uncertainty is

UF ±(B F + t 95SF)

±(18.1 + 2.00 x 37.8)

±93.71b

3.4 END-TO-END CALIBRATION

57, 2.00(III-51 )

A working standard which typically ~~nsists of a strain-gage load cell(s) with aprecise read-out device may be ,obtained. The working standard is installed in the forcemeasurement system such that the force app~~d to the measurement system will also beapplied to the working standard (Fig. 111-7). All electrical and mechanical serviceconnections to the force measuring system should be intact, and all system pressuresshould be adjusted to the nominal levels incurred during engine testing. Care must betaken to ensure that those fluid systems which are pressurized but not flowing duringcalibration, especially the engine fuel system, are designed so that no fluid momentsforces are applied to the thrust stand during testing with fluid flowing. Calibration isaccomplished by applying several known forces in succession (usually eleven' or more) tothe measurement system and recording the results on a digital recording system. The datarecorded are pounds of applied force as indicated by both the working standard and theresulting voltage output of the measurement transducer. Generally, the calibration cyclewill be repeated one or more times'.

54

AEDC-TR-73-5

Typical calibration data for a force measuring system used in engine sea-level statictesting is as shown in Fig. 111.;.15. During the course of sea-level static testing, the. forcesare always applied in one direction only.

Typical calibration data for a force measuring system used in engine altitude testingis as shown in Fig. 111-16. During the course of altitude testing, the resulting forcesexperienced may be either "positive" or "negative" depending on the simulated flight andaltitude conditions.

+

FapPlied

+Fmeasured

+

Fapplied

Fig. 111-15 Typical Calibration Data fromForce Measuring System Usedin Engine Sea-level Testing

+Fmeasured

Fig. 111-16 Typical Calibration Data fromForce Measuring System Usedin Engine Altitude Testing

A force calibration program is utilized in a digital computer to fit a least squarescurve through the calibration data in Figs. III-IS and 111-16. The curve-fit producescoefficients from which measured force may be determined. In practice, curve-fits areusually linear or second order. Occasionally, a third-order curve-fit may be used torepresent a calibration. Higher order curve-fits should not be used without closeexamination. Without a keen knowledge of the relationship between data and a higherorder curve-fit, unrepresentative values may result at interpolated points.

3.5 SUMMARY

Errors from calibration, data acquisition and data reduction processes must beevaluated by one of two methods:

1. Performance of end-to-end calibrations and pre- or post-run appliedload tests

2. Evaluation of all elemental errors

55

AEDC-TR-73-5

In either case, the evaluated errors are combined by the methods outlined in Section3.3, to determine the precision index (SF), bias (BF), and uncertainty (UF) formeasured force.

56

AE DC-T R-73-5

SECTION IVFUEL FLOW MEASUREMENT

4.1 GENERAL

Fuel flow measurements are difficult because there is no standard for measuringvolume per unit of time; therefore, the flow calibrations must be referenced to standardsfor weight or volume. Furthermore, there is no universal method for calibrating aflowmeter.

Turbine meters are the most widely used instruments for measuring fuel flows. Theygenerate an alternating voltage with frequency proportional to the volumetric flow rate.The frequency of the output is converted to an analog voltage and then to digital counts.Another method for recording turbine meter signals is to count the voltage excursions(pulses) over some preset period of time to determine the signal frequency. Figure IV-Iillustrates a typical turbine meter signal.

ten

-+JI""'l 0 I----#---'----I----\---+---+---t---"\---t---o>

oTime .....IIIIr....

Fig. IV-1 Turbine Meter Signal

Turbine meters may be calibrated by three methods:

1. Volumetric: flowIng a measured volume of fluid through the meter toestablish a pulses-per-gallon factor,

2. Gravimetric: flowing a measured mass of fluid through the meter,determining the density, and converting the mass to volume to establish apulses-per-gallon factor, and

3. Comparative calibration: comparing the meter against a master meter.

SPor a defInition of terms used in this Handbook, see the Glossary in Section IX.

57

Calibration factors are determined over a range of turbine meter frequencies todevelop a calibration curve. Figure IV-2 is an illustration of a typical turbine metercalibration curve.

Typically ±O. 5%

f

Meter ~requency, pUlses/secl

Fig. IV-2 Turbine MeteriCalibration l Curve

The range of calibration factors, over the linear range of a turbine meter, is typically±O.5 percent. However, meter range may vary somewhat and is influenced by meter sizeand design, plus fluid viscosity. A complete analysis of turbine meter performance isgiven by' the "Turbine Flowmeter Performance Model." (AD825354) prepared by theGreytad Corporation.

Multiple measurement of fuel flow is recommended for several reasons, the chiefones being reliability and accuracy. Multiple measurements are readily accomplished bysimply installing two or more turbine meters in series. The meters should haveindependent calibrations as far back in the calibration hierarchy as possible.

4.2 FUEL FLOW MEASUREMENT ERROR SOURCES

Errors in fuel flow measurement fall into three major categories: (l) calibration, (2)data acquisition, and (3) data reduction. The calibration elemental bias and precisionerror sources will vary according to the method of calibration used, :while the dataacquisition and reduction errors will not. The next section will contain three parts, onefor each method of calibration. All three may not apply to a particular test facility. Theones that do not should be ignored. The remainder of the section will be devoted todiscussions of the elemental errors in the data acquisition and data reduction processes,

4.2.1 Calibration Errors

Turbine meters are calibrated by three methods or a combination thereof. The firstis volumetric calibration. It is accomplished by flowing a measured volume of fluidthrough the meter and recording the total number of turbine meter cycles (pulses)generated. The calibration factor (K factor) is then calculated

K =to tal pulses

total gallons

58

pulses

gallon(IV-I)

The second method, gravimetric calibration, is accomplished by flowing a measuredmass of fluid through the meter and again recording the total number of turbine meterpulses. Measurement of fluid temperature and pressure will allow determination of fluiddensity. With these data, a meter K factor may be calculated:

K =total pulses pounds-----"-- Xtotal pounds gallon

pulses

gallon(lV-2)

where density = pounds per gallon.

The third method is comparative calibration. The meter being calibrated is comparedagainst a working standard turbine meter. In some applications turbine meters are sorepeatable that the greater part of any precision error incurred results fromnonrepeatability of the calibrating device. In this case, better calibration results can beobtained by simply flowing the calibration fluid through a standard turbine meter inseries with the meter being calibrated. Data recorded are frequency of each meter and theratio (R) of total pulses from the meter being calibrated to total pulses from the standardmeter. The calibration factor (Ke at) for the meter being calibrated is

K pulses = (R)KCal gallon standard

pulseswhere Kstandard = the gallon factor for the standard meter at the set frequency.

(IV-3)

4.2.1.1 Volumetric Calibration

At the apex of the calibration hierarchy, Fig. IV-3,for volumetric calibrations is the National Bureau ofStandards Dynamic Weigh Calibrator. The InterlabStandard Turbine Meter is calibrated against thiscalibrator and in turn is used to calibrate the workingstandard volumetric calibrator in the companylaboratory.

4.2.1.1.1 Calibration of the Interlab Standard

Beginning with the interlab standard, calibrationmethods and elemental error evaluation techniques ateach level in the hierarchy will be discussed.

Turbine meters are calibrated at the NBS asfollows:

59

NBSDynamic Weigh Calibrator

IInterlab Standard

Turbine Meter

IWorking Standard

Volumetric Calibrator

IMeasurement Transducer

Turbine Meter

Fig. IV-3 Turbine MeterVolumetricCalibrationHierarachy

AE DC-T R-73-5

1. At each pulse frequency, specified by the owner, and controlled to withinone-half percent, the total number of pulses generated are counted duringthe period required to flow a weighed quantity of liquid through themeter.

2. Liquid density is determined using measured values of liquid temperatureand pressure.

3. The weighed quantity of fluid is converted to gallons using the densitydetermined in step (2).

4. A pulses/gallon (K) factor is calculated from the data obtained in steps(1), (2), and (3).

S. Steps (l) through (4) are repeated five times successively on each of twodifferent days making a total of ten (10) separate observations.

The K factor reported is the arithmetic mean K of the ten observations. Data from manysimilar calibrations yield a standard deviation (s) of about 0.03 percent for the NBScalibration procedure. The precision index (siC) for the mean value, is then

s 0.03%si( = -- = -- = 0.01%

-IN: y'1O(IV-4)

where Ni = the number of observations in the determination of K. Ni usually is 10; the·degrees of freedom is Ni - I or usually 9. Of course, this is the precision index at just onepulse frequency. The precision indices sj( i for M frequencies must be pooled to producethe prepision index (SN B s) for the calibration process:

±

MI, (N . - 1) s- 2

5=1 1 K i

MI, (N. - 1)i=1 1

(IV-S)

Mwith degrees of freedom dfN BS = .~ (Ni - 1). If the number of tests at each frequency is

1= 110, then these calculations reduce to

M 2I, s- (IV-6)i=1 K i

sNBS ±M

and

dfNBS M(n-I) 9M (IV-7)

60

AEDC·TR·73·5

Bias limits in the NBS dynamic weight calibrator have been established by repeatedcomparison calibrations of reference turbine meters on both volumetric (stand pipe)calibrators and gravimetric (weigh) calibrators. These comparison calibrations wereperformed at several different locations and yielded a bias limit of ±O.l percent. Thisthen is the bias limit (bN B s) reported by the NBS for turbine meter calibration.

Uncertainty in NBS turbine meter calibrations, is then

(lV-8)

If, for example, the number of frequency settings is equal to ten, the degrees of freedomare

dfNBS = 9 x 10. = 90

Since the degree of freedom is greater than thirty,

t95 = 2.00

and

Fig. IV-4 Volumetric Calibrator

U = ±(0.1 + 2.00 x 0.01)

4.2.1.1.2 Uncertainty in the Working Standard

The working standard volumetric calibratorarrangement with liquid level sensors. These markvolume interval (Fig. IV-4). The interlabstandard flowmeter is connected inseries with the calibrator.

To calibrate, liquid is forced out ofor into the calibrator at a constant flowrate. The number of pulses generated bythe flowmeter are counted while theliquid between sensors A and B is beingdisplaced. The volume (Vi) between Aand B is then the quotient of thenumber of pulses counted (Ci) dividedby the turbine meter K factor (KNB S )

determined at NBS:

b.12%

generally consists of a standpipethe top and bottom of a constant

Liquid LevelSensor A

Liquid LevelSensor B

TurbineMeter

Ci

pulses

V. = = gallons1 K NB S pulses/ gallon (lV-9)

Repeating the calibration process N times improves the estimate of the volume, and theaverage of the N calibrations is

NI. v.i=l 1

N

61

(lV-l 0)

AEDC·TR·73·5

The precision index estimate for this determination of volume is the precision index forthe N determinations divided by the square root of N.

Sv = yN

_ N _ 2L (V. - V)i=l 1

N (N-l)(IV-II)

This is called the standard error of the mean for the N determinations. The associateddegrees of freedom is N - 1. Please note in this situation, the precision index of anaverage value (V) is of interest rather than the precision index of the individualdeterminations (Vi). The estimating formula is corrected to provide that estimate bydividing by the square root of N, the number of determinations.

The bias limit (bv) for this process is the root-sum-square of the bias limit reportedby the NBS (bN Bs) and the best estimate of any additional bias contributed by thecalibration process.

Another way to evaluate the precIsIon error of the calibrator is to determine thestandpipe volume by some other means and then compare K factors determined by thecalibrator with those produced by the NBS calibrator. The standpipe volume between Aand B may be determined by physical measurement of the standpipe dimensions or bymeasuring the volume of liquid between A and B with fixed volume standards, e.g., 5-,10-, 50-, I DO-gal containers. Then calibrations of the interlab standard turbine meter canbe performed with the calibrator by forcing liquid into or out of the standpipe throughthe turbine meter at a constant flow rate. The total number of pulses generated whileflowing the volume of liquid between A and B divided by the measured volume yieldsthe meter K factor:

K = total pulses = pulses

total gallons gallon(IV-I 2)

By repeating this procedure several times ·and calculating a mean calibration factor(K) for a constant flow rate as the average of the observed K factors,

where

NLK.i=l 1

N

N = the number of observations used to determine K

Ki = K calculated from the i th observation

(IV-I 3)

and the precision index of K is estimated from the variation of I<:i about K;" it is theprecision index for K divided by the square root of the number of observations (N): "

62

N _ 2L (K. - K)i=l 1

N (N-l)

(IV-I 4)

AEDC·TR·73·5

The degrees of freedom associated with Sl( is N - 1.

If an average K factor (K) is determined at several different flow rates over therange of the meter and corresponding precision index. Sj(, is calculated at each flowrate; then pooling the precision indices provides an estimate of overall calibrator precisionindex:

±

M 2~ (N. - l)SK

j=1 J j

M~ (N. - 1)

J=1 J

(IV-IS)

The degrees of freedom associated with the pooled preCISIon index (the measurementinstrument calibration process) are the sum of the degrees of freedom for each flow rate:

Mdf = ~ (N. - 1)

ws j=l J(lV-l 6)

where M = the number of pulse frequencies for which a K is determined and Nj = thenumber of observations made at each frequency setting. The bias in this process can beestimated as follows:

1. Calculate an average calibration facfor K from Kj values calculated fromthe NBS calibration data

2. Calculate an average calibration factor K from all of the K's calculatedfrom volumetric calibrator data

3. Make the correction K - K

4. Estimates of the limits of unknown bias (bj) for the process should bebased on -data from interlaboratory or interfacility comparisons andengineering judgment. Then

(lV-l 7)

Finally, perform the calibration of the measurement turbine meter against the volumetriccalibrator in essentially the same manner that the interlab standard turbine meter wascalibrated. The precision index (Sj) for this calibration is calculated using Eqs. (IV-IS)and (lV-l 6). The bias limit (b) for the measurement meter calibration process is simplythe best estimate based on interlab or interfacility comparison history and engineeringexperience.

The precision index for the calibration hierarchy is

51 = ±~7Si2

63

(IV-I 8)

AEDC-TR-73-5

51 ±JsNBS2 + s_2 + 2 (IV-I 9)s.v ]

or

51 = ±jsNBS2

+ s 2 + 2(IV-20)s·ws ]

Degrees of freedom (df) for the calibration hierarchy are calculated by

(SNBS2

+ s_2 + Sj 2) 2df1

v (IV-21)4 s_4 s.4

sNBS V J

dfNBS

+ df- + -df.v l

or

(SNBS2

+2 + Sj2)2S

df1ws

SNBS4 S 4 S.4

ws J--+df + df.df

NBS ws J

Bias limits for the hierarchy are

B1 = j,,£ b. 2• 11

B1 JbNBS2 + b-2 + b. 2

V ]

or

B1 = JbNBS2 + b 2 + b. 2

ws ]

Uncertainty in the calibration hierarchy is calculated by

t95 is evaluated at dfl degrees of freedom.

4.2.1.2 Gravimetric Calibrations

The gravimetric flow calibration system has a calibrationhierarchy (Fig. IV-5) similar to that for volumetriccalibrations. The only difference is that the working standardis a dynamic weigh calibrator rather than a volumetriccalibrator.

Fig. IV·5 Turbine Meter GravimetricCalibration Hierarchy

64

(IV-22)

(IV-23)

(IV-24)

(IV-25)

(IV-26)


Interlab StandardTurbine Meter

Working StandardDynamic Weigh Calibrator

Measurement TransducerTurbine Meter

AEDC-TR·73·5

Elemental errors in the gravimetric calibration hierarchy are evaluated in exactly thesame way as those for the volumetric hierarchy except for the working standard.

Turbine meter gravimetric calibrations are accomplished by flowing a weighedquantity of liquid through the meter. Liquid temperature and pressure are measured todetermine liquid density for conversion of the, weighed quantity to gallons. The K factoris calculated by dividing the total number of pulses recorded by the weighed quantity inpounds and multiplying by the density in pounds per gallon:

K = total pulses X pounds = pulsestotal pounds gallon gallon

Figure IV-6 illustrates the basic idea of gravimetric calibration.

(IV-27)

+--Flow

KnownWeights

BalanceMechanism

-:===========tDl========Turbine

Meter

Weigh'------,._..... Ta nk

Fig. IV-6 Gravimetric Calibrator

By flowing liquid at a constant flow rate through the turbine meter and into theweigh tank until the weight of the liquid exactly balances the standard weights, theweighed quantity (W) is established in pounds. With the total number of pulses generated(C) and liquid density (p) in pounds per gallon, Ki is calculated as follows:

K. = pulses = .£1 gallon W P (IV-28)

If Nj observations of Ki are made at one flow rate, average (K) and precision index (SKj)can be calculated:

N.J

~ K.i=l 1

K = N.J

(lV-29)

N.J _ 2~ (K. - K)i=l 1 (lV-30)

65

AE DC-TR-73-5

Then if l( is established for M different flow rates, the precision index for the calibrationprocess is the pooled precision index (Sw s) of the Sj( j indices:

JM 2I (N.- l)Sj(j=l J j

S = ±ws M

I (N. - 1)j=l J

(IV-31)

Degrees of freedom (dfws ) for this process are calculated by

Md£ = I (N. - 1)

ws j= 1 J(IV-32)

Bias in this calibration process is estimated by the method described in Section 4.2.1.1.2.

Calibration hierarchy precision index (Sl), degrees of freedom (df1 ), bias (Bl), anduncertainty (Ul) are calculated by Eqs. (IV-18) through (IV-26).


4.2.1.3 Calibration by Comparison

The comparison method has not been widelyaccepted. However, it does have considerable meritespecially in hydrocarbon fuel applications. The NBS hasrecognized the use of turbine meters as transfer standardsbecause they are very repeatable. The NBS has, in fact,used turbine meters as transfer standards in evaluating biasin the NBS dynamic weigh calibrator.

The third method of calibration is comparison of themeasurement meter with a working standard turbinemeter. This method substitutes a turbine meter for thevolumetric or gravimetric working standards in thecalibration hierarchy (Fig. IV-7).

Working StandardTurbine Meter

Interlab StandardTurbine Meter

Measurement TransducerTurbine Meter

Fig. IV-7 Turbine MeterComparison Calibration Hierarchy

The precision index and bias limit estimates for the interlab standard are made inthe same way as for the volumetric calibration system. The working standard turbinemeter is calibrated against the interlab standard turbine meter by installing the twometers in series in a flow system. Figure IV-8 illustrates a typical setup for comparingturbine meter with turbine meter.

Frequency Frequency

B

Fig. IV-8 ComparativeCalibration

66

AEDC-TR-73-5

By considering, for example, meter A to be an interlab standard and meter B to bea working standard, a calibration can be performed as follows:

1. Adjust the flow rate by observing the frequency of meter A. From theNBS calibration curve, determine the interlab standard K factor for theparticular frequency.

2. With the flow rate adjusted and constant, allow the ratio counter to countthe total number of pulses generated by meter A over some predeterminedtime period. The counter will simultaneously count the total number ofpulses generated by meter B over the same time period. The counter willthen display digitally the ratio (R):

then

and

R = total pulses - A

total pulses - B (lV-33)

(lV-34)

(lV-35)

Thus, calculate Ki for each frequency setting fA. If Ni observations are made at eachfrequency setting, an average calibration factor (K) can be calculated for meter B at fBby

N.J

~ K. (lV-36)K

i=l 1

N.J

The precIsIon index for this average calibration factor is the preCISIon index of thecalibration values divided by the square root of the number of observations (Nj):

N.J _ 2

~ (K B - K)i=l i

N .(N .-- 1)J J

(lV-37)

If a K is determined at M different frequencies over the range of the meter, the precisionindex (sw s) for the calibration process is the pooled value for all frequencies:

M s- 2~ (N. _ 1) Kjj=l J

M~ (N. - 1)j=l J

67

(lV-38)

AEDC-TR-73-5

The degrees of freedom (dfw s) for the calibration process are

Mdf =:£ (N . - 1)

ws j J(lV-39)

Bias limits (bws ) for the process are again based on interlab and interfacility comparisonhistory and engineering judgment.

Precision index (SI ),degrees of freedom (df1 ), bias limits (Bl), and uncertainty(VI) for the calibration hierarchy are calculated using Eqs. (lV-18) through (lV-26).

Some important considerations which have not been mentioned heretofore are:

1. At each level in the calibration hierarchy, the flowmeter being calibratedshould be accompanied by the plumbing upstream and downstream of themeter in its use condition. Tests have shown that inadequate control ofthe velocity profile is perhaps the strongest argument against the use ofreference turbine meters as standards (see page 184 of "TurbineFlowmeter Performance Model," (AD825354) prepared by GreyradCorporation).

2. If at all possible, turbine meters should be calibrated with the use liquid atrun conditions of temperature and pressure.

These considerations will minimize the errors:

4.2.2 Data Acquisition Errors

The effect of data acquisition errors is determined by applying a known frequency(X) to the data acquisition equipment (Fig. IV-9).

Signal Conditioning Electrical FrequencyEquipment Source for Calibration

I It , 1 t

I Oscillograph I Digital AID OtherRecording

Counter Converter Systems

~ ~Digital Recorderor Equivalent

tFrequency Digits Pulses

Fig. IV-9 Data Acquisition System Calibration

68

AEDC-TR-73-5

In the case of digital counters, the total number of input pulses is counted oversome preset period of time and the digital indication is recorded. Then the recorded valuecan be compared with the input. If M recordings on N channels are made, then theaverage (Xj) for one channel is

M X.X. :£ --2.

J i=l M(IV-40)

where Xi = the recorded value for the ith recording. The grand average (X) for allchannels is

x

N:£ X.j=l J

N

(IV-4I)

The precision index (sx) for digital counter channels is

Sf = ±

N _ _ 2:£(X.-X)j=l J

N - 1

(IV-42)

The degrees of freedom (dfj ) are the number of recordings minus one, i.e., N-l.

Bias limits for the data acquisition process are left to the judgment of the mostknowledgeable data recording engineers. Errors incurred by frequency-to-analog andanalog-to-digital conversions are evaluated by comparing known input frequencies (f) withrecprded frequencies. The known frequency (f) is applied to N channels, and M multiplescan recordings are made. A_multiple scan average (fij) is calculated for each channel foreach recording. The average (fj) for each recording on the jth channel is

The grand average (f) for all channels is

r.J

M:£ f ..i=l 1J

M

(lV-43)

The precision index (sf) is then

(IV-44)

N _ _ 2

:£(f.-Oj=l J

N - 1

69

(IV-45)

The -degrees of freedom (dff) in this case are

dfy = N - 1 .(lV-46)

Bias limits (br) for frequency-to-analog and analog-to-digital conversion must be estimatedand root-sum-squared with the bias limits (bro) for the input frequency (f) as determinedfrom the frequency standard calibration hierarchy, i.e.,

4.2.2.1 Multiple Instruments

Because of the lack of flow rate standards, i.e., gallons per minute or pounds persecond standards, multiple instrumentation for flow measurement is highly recommended.Correct application of multiple measurement statistics will never yield an estimate ofprecision error larger than that obtained with single measurements. The typical multipleinstrumentation situation provides a reduction in precision error. The measurementprovided by the average of sevetal instruments is more precise than any individualinstrument of that set. The reduction of precision error is indicated by the precisionindex (savg) of several instruments:

(lV-47)

where Sind = the precIsIon index of the individual instrument and K is the number ofinstruments. The formula for calculating Savg is as follows when the simple average ofmultiple instruments is used:

If individual instruments are weighted when combined, the formula for Sa vg is morecomplex.

Analysis of flowmeter-to-flowmeter multiple measurements yields

1. Pooled within-run precision index (Sw r) and

2. Pooled run-to-run precision index (Srr).

Appendix C gives derivations and formulation for calculating the above precision indices.

Further analysis of m~ltiple measurement data will provide an estimate of flowmetercalibration-to-calibration precision error (See) which includes

1. Flowmeter nonrepeatability during calibration,

2. Installation effects between calibration facility and engine test stand, and

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AEDC-TR-73-5

3. Calibration facility nonrepeatability, which includes temperature andpressure errors in defining density during calibration.

Derivations and formulation for evaluating See are also in Appendix C.

4.2.3 Data Reduction Errors

Data reduction errors are those errors incurred in reducing measured units to unitsof flow and fall into the following three categories:

1. Density determination errors,

2. Precision errors resulting from test dynamics, and

3. Computer resolution errors.

4.2.3.1 Density Determination Errors

Errors in density determination are a result of errors in the measurement of fueltemperatures and pressures. At this point, these errors may be called Sde for the precisionindex and bde for the bias limits.

The effect of fuel pressure precision errors on fuel density is calculated from

where

1 apSd = ±- -Sp

e 1 vIC ap

Sp = Pressure measurement precision index

(lV48)

1yK = Factor to account for K multiple pressure measurements

ap= The partial derivative of the fuel density versus pressureap relationship

The degrees of freedom associated with Sde are the same as that for the pressuremeasurement:

(lV49)

The effect of fuel pressure measurement bias on fuel density is

(IV-50)

The effect of fuel temperature precision error on fuel density (Sd e2) is calculated asfollows:

71

AEDC·TR·73·5

(IV-51 )

where ~ = factor to account for K multiple temperature measurements

:~ = the partial derivative of the fuel density versus temperaturerelationship.

The degrees of freedom associated with Sd e 2 are the same as the degrees of freedomfor the temperature measurement precision index, i.e.,

(IV-52)

The effect of fuel temperature bias on fuel density (Bde2 ) is calculated from

(IV-53)

4.2.3.2 Computer Resolution

Computer resolution is the source of a small elemental error. Even the smallcomputers used in experimental test applications have six digit resolution (most haveeight or more). The resultant full-scale error would be plus or minus one in 106 . Eventhough this error is probably negligible, some consideration should be given to it.

Two types of measurement resolution systems are in use, truncating systems androunding systems. Consideration will be given the elemental bias and precision errorsinherent in each of these.

In multiplYing or dividing with six or eight digit numbers, the results may have moredigits than the system resolution. The truncating system will eliminate digits on the rightuntil the maximum allowable are left. This results in a bias of 1/2 digit and a uniformdistribution of precision errors over the interval ±1/2 digit. The elemental precision indexfor this type of distribution is derived from the precision index of a uniform distribution-(Appendix A):

s(lower limit - upper limit) 2 .... [';2

12 = -,12 = to.3 (IV-54)

The elemental precision error will be ±3/10 of a digit.

The rounding system will also reduce the number of digits to the resolution of thecomputer. In doing this it will increase the first digit on the right by one approximatelyhalf of the time. The decision to increase the digit is based on the size of the last digiteliminated (and others if necessary).

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AEDC·TR·73-5

The bias error of 1/2 digit experienced with the truncating system is eliminated. Theprecision error, however, is not eliminated. It remains the same, ±3/10 of a digit. For asix-digit computer, the precision error is

SCR = ±3/10 x 10-6

The conclusion is that the rounding system is superior to the truncating system, andif a choice exists, the rounding system should be used.

4.3 FUEL FLOW MEASUREMENT ERRORS

Fuel flow measurement errors when flowmeter calibrations are performed off siteand multiple measurements are made, are then

1..1 2 2 2 252 25 = ±vI( ., 51 + Sx + Swr + 5 rr + cc + sde (IV-55)

where S = precision index for the fuel flow measurement and~ = the factor to account

for K multiple instruments which are averaged to provide the measurement.

(5 12 + 8- 2 + S 2 + S 2 + S 2 + 2)2

dfx wr rr cc Sde

(IV-56)S 4 8- 4 4 4 4 4S S S sde1 X wr --!:.!:- --.£.£..dT + df- + df + df + df + dfde1 X wr rr cc

where df = degrees of freedom associated with the precision index (S). The bias limitusually is not reduced when multiple instruments are used because they usually haveequal biases. Bias errors would be reduced by averaging if the meters were calibratedindependently at different facilities. The bias limit would be

(IV-57)

where B = the bias limits for the measurement process. In Eqs. (IV-55) through (IV-57)if a frequency-to-digital conversion process was employed rather than counters, S5(, bx,and dfx may be replaced with Sf, bf, and dff. The uncertainty would be

(IV-58)

4.4 END-TO-END CALIBRATION

The best method {or determining the errors incurred in the flow measurementprocess would be to flow a weighed quantity or measured volume of fuel through theturbine meters during an engine run. That is, an in-place, end-to-end calibration would beperformed during an engine run. Figure IV-10 illustrates a typical arrangement ofcomponents for performing calibrations of this nature.

73

AEDC-TR-73-5

As an example, suppose a measured quantity (gravimetrically or volumetrically) offuel was flowed through two flowmeters in series to the gas turbine engine during. a run.During the period required to flow the measured quantity, record the following:

1. Total pulses generated by each meter,

2. Fuel temperature and pressure for density determination, and

3. The value for the measured quantity in gallons or pounds.

Fuel Fluid

Supply ...... ------- CalibrationSystem

I

"Flowmeter Gas Turbine

Measurement Engine

ElectricalFrequency

Calibration

Signalr-- Conditioning

Equipment !• Other

IOSCillOgraPhl IDigital .1 1 AID I RecordingCounter Conversion Systems

• IDigitalRecorder orEquivalent

Icalibration~I Factors

DataProcessingEquipment

~Frequency Flow Rate

Fig. IV-10 End-to-End Calibration

Flow Rate

From these data a meter factor can be calculated.

K. = total pulses1 total gallons

(IV-59)

for each meter. Then if N runs are made the precision index for data acquisition andreduction for each meter is

S.1

N _ 2I (K. - K)i=l 1

N - 1

(IV-60)

74

(lV-6l)

, AEDC-TR-73-5

This precision index will include the effects of

1. Meter nonlinearity

2. Electrical calibration

3. Counter or other frequency-to-digital conversion

4. Digital recording

S. Density determination


7. Calibration to calibration precision error

The only precision errors not included in Si are within-run-precision error (Sw r) and theprecision error (sws) of the working standard (calibration system) calibration process.Fuel flow measurement precision index (S) is then

1 _, 2 2 2S = ±- "'l s. + S + swsyK 1 wr

where Jr<. = the factor to account for averaging K multiple instruments. Derivations and

formulation for calculating Sw r are in Appendix C. The working standard precision errorcan be determined using Eq. (IV-IS) or (lV-30), depending on the type of workingstandard used.

Degrees of freedom for the flow measurement process are

2 2 s 2)2

(Si + S +df = wr ws

(lV-62)4' 4 4s. S S

1 wr wsK+ ~+ df

1 wr ws

where df defines individual degrees of freedom associated with each precision index.

Bias limits (B) in the flow measurement made by this method are the bias in theworking standard plus any additional bias based on engineering judgment.

Uncertainty in the measurement is then

(lV-63)

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AEDC-TR-73-5

4.5 SUMMARY

In summary, the elementalerrors to be evaluated are listed inTable XIII when off-site calibrations are performed. If end-to-endcalibrations are performed while anengine test is in progress, theagony of evaluating all of 'theabove errors is eliminated exceptthose errors determined from multiple measurement statistics whichmust be evaluated in either case.

Table XIII Elemental Errors

Bias Precision Degrees ofSource Limit Index Freedom

Calibration Bl SI dfl

Data Acquisition bx Sx, Swr, dfx, dfwr ,Srr, Sec dfrr , dfee

Data Reduction bde sde dfde

76

AE DC-T R·73-5

SECTION VPRESSURE AND TEMPERATURE MEASUREMENTS

5.1 GENERAL

Pressure and temperature measurements are important in the evaluation of gasturbine engine performance parameters6 , and this section is devoted to the considerationof errors in measurements.

Fig. V-1 Strain-Gage Pressure Transducer Circuitry

ExcitationVoltage Source

Strain Gages

Strain-gage-type pressuretransducers are the mostcommon devices used for themeasurement of pressures ingas turbine engine testing.Strain-gage pressure transducers are best described asdevices which have straingages fixed directly to adiaphragm or other pressuresensitive surface. The measured strain is calibrated interms of pressure. Figure V-Iis a simplified diagram illustrating the circuitry for measurement of pressure with a strain-gage-type pressure transducer.

Output Voltage

II

Transducer • L• Pressure~ Source

PressureStandard

IAppliedPressureRecorder

There are two different methods fOF calibrating pressure transducers. One method (Fig. V-2)consists of subjecting the transducer-and a pressure standard to the samepressure and recording the outputs.Eleven or more pressure levels overthe range of the transducer arerecommended. One such pressurestandard, which has gained in popularity in recent years, employs afused quartz helical bourdon tube andoptical techniques to produce adigital indication of the appliedpressure.

Fig. V-2 Pressure Transducer Calibration withPressure Standard

6Por a definition of terms used in this Handbook, see Glossary in Section IX.

77

AE DC-T R-73-5

Transducer

WeightsThe second method (Fig. V-3)

consists of applying known pressuresto the transducer using an air orliquid deadweight piston gage.

Thermocouples are the mostcommonly used devices for measuringgas turbine engine temperatures. Resistance thermometers are also used ingas turbine testing. Figure V-4 illustrates typical hook-ups for temperature measurement using resistancethermometers and thermocouples.

Fig. V-3 Deadweight Piston Gage There are two methods of cali-brating thermocouples and resistance

thermometers: (l) by comparison with fixed temperature points and interpolated pointsas defined by the International Practical Temperature Scale of 1968, and (2) bycomparison with platinum-rhodium thermocouples or resistance thermometers used asstandards.

Excitation

BridgeCompletion

Network

a. Resistance ThermometerThree-Wire System

\

MeaSUrin gI Junction

I

i ~Metal A

l---LMetal BI

ReferenceJunctions

I--->CopperWire

c ~

VoltageMeasuringInstrument

b. Thermocoupie System

Fig. V-4 Temperature Measurement

Multiple concurrent measurements of pressure or temperature parameters along thegas path of a gas turbine engine are rarely achieved because of profile effects. Multipleconcurrent measurements of pressure or temperature parameters in a turbine engine fuelsupply system or airflow measuring system, however, are easily accomplished and shouldbe practiced whenever accuracy requirements dictate.

78

AE DC-T R-73-5

Fig. V-5 Probe Boss Arrangement

As an example, multiplemeasurement of fuel temperature is accomplished by theinstallation of transducers inadjacent probe bosses in a man- a. Front Viewner which will allow the twotransducers to see the sametemperature. Figure V-5 illustrates an accepted method of probe boss arrangement.

....--FluidStream

b. Side View

When the average pressure at a given turbine engine station is required, severalindividual pressure measurements, each at a unique geometric location, are generallymade. If the individual pressure measurements are independent, then the precision indexof the average pressure is

(V-I)

where Sp is the probe precision index, N is the number of independent pressuremeasurements, and ST is the transducer precision index.

5.2 PRESSURE MEASUREMENT ERROR SOURCES

5.2.1 Calibration Hierarchy Errors

Figure V-6 illustrates a typical pressure calibrationhierarchy.

In the typical hierarchy, deadweight piston gages areused as standards at the upper three levels. Calibration ofdeadweight piston gages is accomplished by comparisonswith other deadweight piston gages. Figure V-7 illustratesthe setup for this method of calibration.

Weights are applied to the weight table of testernumber one until the weights on tester number two are

Weights

NBSDeadweight Piston Gage

and Standard WeightsI

Interlab StandardDeadweight Piston Gage

and Standard Weights

Reference StandardDeadweight Piston Gage

and Standard WeightsI

Measurement InstrumentStrain-Gage Transducer

Fig. V-6 Pressure Transducer Calibration Hierarchy

Fig. V-7 DeadweightPiston GageCalibration

79

AEDC-TR-73-5

exactly balanced. Knowledge of the effective area of the piston in tester number one andthe total mass of the applied weights, weight table, and piston for each tester will allowcalculation of the effective area of the piston in tester number two by the relationship

(V-2)

then,

(V-3)

where Ai = the effective area of the piston in tester number one in squareinches

A2 = the effective area of the piston in tester number two in squareinches

Mi = the total mass of applied weights, weight table, and piston fortester number one in pounds

M2 = the total mass of applied weights, weight table, and piston fortester number two in pounds

Determination of the piston area at several -pressures over the range of the gage andrepeatedly at each pressure point will yield a distribution of data. The precision index(SA) is the standard error of the mean for the calibration process anj is estimated fromthe variation of the individual area determinations about the average (A). When

N~ A.

A. j=l ]

1 N

and

K~ A.

Ai=l 1

K

thus,

(V-4)

(V-S)

sA

N _ 2~ (A. - A.)j=l J 1

K (N - 1)

(V-6)

80

A EDC-T R-73-5

where N = the total number of observations made at each pressure set pointduring the calibration

Ai = the average piston area at each pressure set point

A = the average effective piston area

K = the number of pressure set points

Aj = the area determination at each individual observation at each setpoint

Degrees of freedom (dfA:) associated with the precision index (SA:) are

dfA = K (N - 1) (V-7)

(V-8)[l+a(t-t)][l+bp][l+d(p +Sp -p.)]

s p zo Z P J

Pressure measurements can be made with deadweight piston gages with uncertaintiesof ±0.01 percent. To do so, a number of parameters of the instrument and itsenvironment must be considered. Consideration of these parameters is primarily for thepurpose of minimizing the bias errors incurred by variations in the parameters. Principalinstrument parameters are evaluated during calibration, but the user must evaluate thoseof the environment. For example, the pressure developed by a deadweight piston gage isgiven by the formula

M m ( Pa) V(Pfa - Pa) yC- 1 - - kg + kg +-Ao Pm L A

oL A

o

where Pp =pressure at the reference level, psi

Mm =mass of the weights, including the piston assembly, lb

Ao =effective area (mean area of the piston and cylinder) inin. 2 , at atmospheric pressure, temperature ts , andjacket pressure pzo

Pa = mean density of the air displaced by the load, Ib/in.3

Pm =density of the weights, Ib/in. 3

k = 1/980.665

gL =local acceleration due to gravity, cm/sec2

V = volume of oil in in. 3 contributing to the load on the piston

Pfa =density of the pressure fluid at atmospheric pressure, Ib/in.3

'Y = surface tension of the pressure fluid, lbf/in.

C =circumference of the piston assembly in inches at the surfaceof the pressure fluid

81

AEDC-TR-73-5

a =fractional change in area per 0 C, and taken as equal to the sumof the thermal coefficients of linear expansion of the pistonand cylinder

t =temperature of the piston gage, 0 C

ts = reference temperature at which the value for Ao is known

b = fractional change in effective area for I-psi change in pressure

d =fractional change in effective area for I-psi change injacket pressure

pz 0 = jacket pressure in psi required to reduce the piston-cylinderclearance to zero when Pp = 0

Sz =rate of change of zero clearance jacket pressure (Pzo) withmeasured pressure (pp), psi

Pj = jacket pressure, psi

Obviously, treatment of each of these variables is beyond the scope of this work. They arepresented so that the reader will be aware of the considerations that must be made if bias isto be held to a minimum. The NBS will report bias limits for any calibration performed atthe Bureau. Bias limits at the lower levels are dependent on the amount of calibration dataavailable and thejjudgment of the one responsible for calibration data analysis. The bias (bi)at each level in the hierarchy is at least as large as the bias estimated for the previous level;therefore a conservative estimate is

(V-9)

Upon completion of a strain-gage pressure transducer calibration, a least-squarespolynomial curve may be fitted through the data such that

where P = pressure in psia, psid, or psig

Ao, AI, A2 , and AM = curve coefficients

X = transducer output voltage

K = degree of the curve fit, i.e., thelargest exponent of P

M = the number of curve coefficients

(V-I 0)

The errors attributed to the calibration process are then the same as those discussed inSection III for laboratory-calibrated force transducers. The precision index (Si) at anypressure Pi is calculated (Fig. V-8) as

S.1

±

N 2~ (X. - X)

i= 1 1

N - 1

82

(V-II)

AEDC-TR-73-5

where Xi =measured output voltage corresponding tothe applied pressure (P)

X =output voltage from the curve at pressure (P)

N =number of calibration points at pressure (P)

The degrees of freedom (dfi) associated with this precision index are

The best procedure is to calculate aprecision index for each pressure set pointand report the largest as the precision indexfor the calibration process.

N (x - X) 2L i

i-I N - 1

x .. f(Ai, Pj)Where i - 0 to HAnd j" 1 to K

P(V-12)dfi = N - 1

Bias limits for this calibration processare left to the judgment of the instrumentation engineer.

Applied Pressure~

Fig. V-8 Precision Index at Any AppliedPressure (P)

The precision index for the complete hierarchy is calculated by root-sum-squaring theSi for each level:

(V-13)

The degrees of freedom for the complete hierarchy are estimated using theWelch-Satterthwaite technique:

(V-14)

The bias limit for the hierarchy would be the root-sum-square of the bias limits foreach level:

(V-IS)


One. method of determining the overall effect of the data acquisition and reductionprocesses is by the periodic performance of special tests. First, select several (four to ten)pressure transducers at random from the pool of calibrated pressure transducers. Cqnnectthe transducers to a common manifold (Fig. V-9). Follow the customary recording systempre-run setup procedures. Apply nominal pressure to the manifold, and make a multiplescan (ten or more) recording of transducer outputs. Also, record the level of the appliedpressure as defined by a pressure standard (deadweight piston gage or equivalent). Return

83

AE DC-TR -73-5

Transducers

PressureSo'urce

Manifold

Fig. V-9 Periodic Pressure Tests

(V-I 6)X..1J

the manifold pressure to atmospheric, and then again apply the nominal pressure and recordas before. Repeat this procedure at least four times, i.e., make at least four multiple scanrecordings of transducer outputs with nominal pressure applied. Reduce the data by meansof the normally used gas turbine engine data reduction program. Calculate a multiple scanaverage (Xij) for each transducer at each set point:

NI X "k

k= 1 I)

N

where N = number of scans recorded and Xijk = pressure indicated by the jth transducer onthe ith scan of the kth recording. The average (Xj) for all set points on the jth transducer is

X.J

MI x..i=l I)

M

(V-I?)

where M = the number of pressure set points.

The precision error for the dat~acquisition and reduction system is estimated by thevariation of the set point recordings (Xij) about the average (Xj) for each transducer:

M.J _ _ 2

I (X .. -x.)i=l IJ )

Sj = M. _ 1)

(V-I8)

This estimate is pooled for L transducers to estimate the precision index for the dataacquisition and reduction system:

s-x

LI (M. _ 1)8. 2

j=l) )

LI (M. - 1)j=l )

L M.) _ _ 2

I I (x .. - x.)j=l i=l I) J

LI (M. - 1)j=l J

(V-I 9)

84

AEDC-TR-73-5

The degrees of freedom (df) associated with the precision index are

Ldf- = L (M. - 1)

x j= 1 J(V-20)

Bias limits for these tests will be based on engineering judgment supported by special testhistories.

These statistics include the effects of the following:

1. Pressure standard

2. Pressure transducer

3. Excitation

4. Electrical calibration of the recording system

5. Signal conditioning

6. Recording system

7. Data reduction

but do not include the effects of

1. Probe or tap design and/or coupling configuration

2. Environmental conditions

3. Pressure variations during engine testing

5.2.3 'Probe Errors

Treatment of errors from probe or tap design is beyond the scope of this Handbook.The reader is referred to several excellent references which should provide the backgroundrequired to complete an error analysis:

1. "Aerodynamic Measurements," Robert C. Dean, Jr., MIT Press, 1953, Chapters 3and 5.

2. "Considerations Entering into the Selection of Probes for Pressure Measurementsin Jet Engines," Clarence C. Gettelman and Lloyd N. Krause, ISA Proceedings,Volume 7, Paper No. 52-12-1.

3. "Effect of Interaction among Probes, Supports, Duct Walls, and Jet Boundarieson Pressure Measurements in Ducts and Jets," Clarence C. Gettelman and LloydN. Krause, ISA Proceedings, Volume 7, Paper No. 52-12-2.

4. "Review of the Pitot Tube," R. G. Folsom, ASME Transactions, October 1956.

85

AE DC-T R·73-5

5. "Characteristics of a Wedge with Various Holder Configurations for StaticPressure Measurements in Subsonic Gas Streams," NACA RM E51G09,September 1951.

6. "An Investigation of the Influence of Orifice Geometry on Static PressureMeasurements," R. E. Rayle, Jr., MS Thesis, MIT, 1949.

As an example of the type of error referred to here, suppose an engineer wants to makevelocity coefficient corrections to total pressure measurements. Very likely, the correctionwill be of the form

where PT = actual total pressure

PT m = measured total pressure

(PTm - PT )/q = velocity coefficient

q =velocity head =~ PV2

(V-21)

Velocity coefficients for each probe may be determined by calibration. Calibrationdata will yield the precision inct'ex (SPT ) and bias limit (BpT ). Expanding Eq. (V-21) inthe Taylor's series gives m m

Sp ± SP Trn2 + WTrnq-PT)sT (V-22)

T

and

(PT-PT)Bp ± Bp T 2 + q2 B2

mq (V-23)T

m

It is to be noted that the term (PT m - PT )/q is treated as a constant with bias erroronly, whereas the term q has only precision error with bias being negligible.

Pressure variations caused by the engine or facility are not errors. The reader iscautioned that time variant data adds additional complexity to the uncertainty analysis (seeSection 8.7).

The pooled within-run precision index (Sw r) and pooled run-to-run precision index(Srr) may be obtained from multiple measurements. Derivations and, formulation forcalculating Sw rand Srr are in Appendix C.

86

(V-24)

AE DC-T R-73-5

5.2.4 Pressure Measurement Error Summary

Finally, if precautions are taken to make environmental effects negligible or laboratorytests are performed similar to those suggested in Section III for determining their effects,the measurement precision index may be calculated from

Sp = ±~S12 + Sj{2 + Swr2 + Srr

2 + se2 + sb

2 + SPT2


Sx = data acquisition and reduction precision index

Sw r = within-run precision index (multiple measurements only)

Se = precision error due to environmental effects

Sb = precision error due to the measurement of barometric pressureif transducer zero corrections are in the data reduction processfor absolute pressure measurement or if gage pressuremeasurements are made

Srr = Run-to-run precision index (multiple measurements only)

SPT = probe velocity coefficient precision index (when applicable)

The degrees of freedom (df) associated with the precision index are,

(S12

+ S_2 + S 2 + S 2 + s 2 + s 2 + S 2)x wr rr e b P Tdfp

S 4 S- 4 S 4 S 4 s 4 s 4 Sp 4 (V-25)1 x wr rr e b T+- +-- + + df + -+

df 1 dfdf df df bdfpx wr rr e T

B~as limits for the pressure measurement are

B p = ±VB12 + B x2 + Be

2 + B b2 + B

PT2 (V-26)

If ~pecial tests are not performed to determine the overall effect of the data acquisitionand reduction processes, then each of the elemental errors must be evaluated as in SectionIII. The precision index for the pressure measurement would, therefore, be calculated asfollows:

where

S S 2 S2 S2 S 2 S2 2 2 S 2P = ± 1 + 2 + 3 + wr + rr + se + sb + P T

S1 = the root-sum-square of the elemental precision errorsin the calibration hierarchy

S2 = the root-sum-square of the elemental precision errorsin the data acquisition process

S3 = the root-sum-square of the elemental precision errorsin the data reduction process

87

(V-27)

AEDC·TR·73·5

The degrees of freedom for the pressure measurement are

(5 12 + S 2 + S 2 + S 2 + S 2 + S 2 + 2)2

dfp2 3 wr rr e sb

(V-28)S 4 S 4 S 4 S 4 S 4 s 4 s 4

1 2 3 wr rr e bdf + ""df + F+ df

+ -- +df

+--elf dfbI 2 3 wr rr e

Bias limits for the pressure measurement are calculated as the root-sum-square of theelemental bias limits

(V-29)

5.3 TEMPERATURE MEASUREMENT ERROR SOURCES

5.3.1 Calibration Hierarchy Errors

The apex of the calibration hierarchy is, of course, the NBS as indicated in Fig. V-IO.

NBSPrimary Standards and

Platinum Resistance Thermometer

IInterlab Standard


ITransfer Standard



Platinum Resistance Thermometeror

Thermocouple

NBSPrimary Standards and

Pt-IO Rh/Pt Thermocouple

IInterlab Standard

Pt-IO Rh/Pt Thermocouple

ITransfer Standard

Thermocouple


Platinum Resistance Thermometeror

Thermocouple

Fig. V-10 Temperature Transducer Calibration Hierarchy

The NBS certifies platinum resistance thermometers by the International PracticalTemperature Scale of 1968 (lPTS-68). The International Practical Temperature Scale isbased on a number of fixed and reproducible equilibrium temperatures and oninternationally agreed-upon measuring instruments and interpolation formulas. The elevenprimary equilibrium temperatures and a few secondary temperatures are listed in Table XIV.

88

AEDC-TR-73-5

Table XIV International Practical Temperature Scale of 1968

Reference Temperature

°c oK of oR

Freezing Point Gold* 1064.43 1337.58 1947.97 2407.64Freezing Point Silver* 961.93 1235.08 1763.47 2223.14Freezing Point Aluminum 660.37 933.52 1220.67 1680.34Freezing Point Antimony 630.74 903.89 1167.33 1627.00Boiling Point Sulfur 444.674 717.824 832.413 1292.083Freezing Point Zinc* 419.58 692.73 787.24 1246.91Freezing Point Lead 327.502 600.652 621.504 1081.174Freezing Point Tin 231.9681 505.1181 449.5426 909.2126Boiling Point Water* 100. 373.15 212. 671.67Triple Point Water* +0.01 273.16 32.02 491.69Boiling Point Oxygen* -182.962 90.188 -297.332 162.338Triple Point Oxygen* -218.789 54.361 -361.820 97.850Boiling Point Neon* -246.048 27.102 -410.886 48.784Boiling Point Hydrogen* -252.87 20.28 -423.166 36.50Boiling Point Hydrogen, -256.108 17.042 -428.994 30.676

25/76 atm*Triple Point Hydrogen* -259.34 13.81 -434.812 24.86

*Indicates IPTS-68 Defining Temperature

A typical calibration report from the NBS reads as follows:

Temperatures between oOe and 630.74°e on the new International Practical TemperatureScale of 1968 (IPTS-68) are defined by the indications (resistance values) of standard platinumresistance thermometers and the following expressions:

+ B(~ - 1)~100 100

(1 )

(2.)

M( t')(

t; J( t; ~ ( t ; ~ ( t ; )o 045 - - - 1 - 1 -- - 1. 100 100 419.58 630.74

(3)

where t is the temperature, at the outside of the tube protecting the platinum resistor, in °e onthe International Practical Temperature Scale of 1968 and Rt and Ro are the resistances of theplatinum resistor at tOe and oOe, respectively, measured with a continuous current through theplatinum resistor. The value of M(t'), given by expression (3), is the same for all thermometersand is a function only of the quantity t'. The addition of the small value represented by (3)serves to make the IPTS-68 conform more closely to the thermodynamic scale than can be donewith only the simple quadratic of expression (2).

An alternate form which is completely equivalent to expression (2) is

(4)

89

AEDC-TR-73-S

In some instances expression (4) is less difficult to calculate than (2). The constants A and Bused in (4) are related directly to a and o.

A

B

a (l + 0/100)

-ao/10 4

(5)

(6)

CAUTION: The values of A, B, and o'on the new 1968 scale are distinctly different fromthe corresponding values on the old 1948 or 1927 scale. The values of a and Ro are alsodifferent but onry trivially so.

Temperatures below OOC on the new 1968 scale are calculated using a standard referencetable which gives values of RtlRo for a fictitious "mean" standard thermometer. This referencetable and a specified deviation equation are combined to give the values for a particularthermometer. The standard reference table used for IPTS-68 is referred to as the "CCT-68"table. It is convenient to use the symbol Wt in place of Rt/Ro . For the special reference valuesof Rt/Ro tabulated in CCT-68, the special symbol Wt* i;U=secr:-The table giving values of Wt fora particular thermometer from OOC down to -182.962°C may be calculated from the followingexpressions:

(7)

(8)

Expression (8) is the specified deviation equation in the range OOC to -182.962°C....

A table calculated from the constants for this thermometer is on the following pages. If novalue for C4 is given, the table below OOC was calculated with an assumed value of thisconstant. The first column of the table gives values of temperature. Unless a different functionis requested, the second column gives FfJRo (i.e. the ratio of the ~sistance at the statedtemperature to the resistance at the ice point). The third column gives the inverse (reciprocal)of the difference between successive values in the second column. These reciprocal firstdifferences are included to facilitate interpolation. The error introduced by using linearinterpolation will be less than O.0001°C.

The range of this table does not imply that this thermometer is necessarily a satisfactoryinstrument over exactly the same range. The range was selected to cover an interval believed toinclude the needs of the majority of users of this type of thermometer.

Temperature transducers certified by this procedure by the NBS exhibit negligibly smallerrors when compared with errors associated with calibration processes below the NBS level.

Calibrations of transfer standard temperature transducers are accomplished by placingthe transfer standard in a calibration medium along with the interlab standard which definesthe temperature of the medium. If comparisons are made between the interlab standard andtransfer standard at K temperatures and the process is repeated at specified time intervals, acalibration data bank will be established from which calibration precision indices and biaslimits may be established, i.e.,

90

s. = ±J

N~ (x .. - X.)i=l 1J J

N - I

AEDC-TR-73-5

(V-30)

where Sj = calibration process precision index at temperature j

Xij = individual temperatures indicated by the transferstandard at temperature j

Xj = the average indication by the transfer standardat temperature j

N = number of indications at temperature j

(V-3l)

where ST S is the calibration process precision index for the transfer standard. Thedegrees of freedom (dfTS ) are

dfTS = K(N - 1) (V-32)

Bias limits for the transfer standard calibration process should be estimated on thebasis of interlab and interfacility comparisons and engineering judgment.

Temperature transducers used in jet engine testing are usually calibrated against atransfer standard transducer. The same techniques described for transfer standards are usedin these calibrations. Comparisons are made at a minimum of three different temperaturesover the test transducer range to establish conformance with the manufacturer'sspecifications. If the manufacturer specifies limits for precision error and bias, the precisionindex for the transducer is the precision error limit divided by t = 2.0, i,e., if

then

Si = ±0.18

(V-33)

(V-33)

where t95 Si is the specified precision error limit and Si is the transducer precision index.

The degrees of freedom (dfi ) associated with the precision index are assumed to begreater than 30.

Bias limits (Bi) for the transducer are equal to the limits specified by the manufacturer.

If extensive calibration histories have been accrued, it may be advantageous tocalculate the precision index directly from calibration data rather than accept the values

91

AEDC-TR-73-5

provided by the manufacturer. The manufacturer's specified limits will likely beconservative. This being the case, the precision index is calculated from

±

N _ 2~(X .. -X.)

i=l 1) )

N - 1(V-34)

where

where

ST j =the precision index for N transducers at temperature j

Xij =individual indication at temperature j by the ith transducer

Xj =the average indication at temperature j by all transducers

N =the number of transducers calibrated

~2

.~ ST.S

)=1)

T=± -M-

ST = measurement transducer precision index

M =number of temperatures at which comparisons are made

(V-35)

The degrees of freedom for this calibration process are

dfT = M(N - 1)

Bias limit estimates are left to the judgment of the instrumentation engineer.

The precision index for the calibration hierarchy is

The degrees of freedom for S1 are

The bias limit for the calibration hierarchy is


(V-36)

, (V-37)

(V-38)

(V-39)

Evaluation of the overall effect of data acquisition and reduction errors is bestaccomplished by monitoring systems or special tests specifically designed for this purpose.

92

AEDC-TR-73-5

5.3.2.1 Thermocouples

A reference temperature monitoring system will provide an excellent source of datafor evaluating errors due to the recording apparatus and data reduction process.

Fig. V-11 Typical Thermocouple Channel

LTRJ

~ Uniform TemperatureReference

rI II Ice I

TO I Point IL..!:!. :....J

Figure V-II depicts a typi-Cr r-,

cal setup for measuring Cr CugasT1 <Al

I :: : Iturbine engine gas path temper- Al I Cu

atures with Chromel®-Alumel® I Ithermocouples.

Cr Cu

Cu

If several calibrated thermocouples are utilized to monitor the temperature of thereference junction block, statistically useful data can be recorded each time test data arerecorded. Assuming that thosethermocouple data are recorded and reduced to engineering units by processes identicalto those employed for engine temperature measurements, a stockpile of data will begathered, from which data acquisition and reduction errors may be estimated.

For the purpose of illustration, suppose N calibrated Chromel-Alumel thermocouplesare employed to monitor the reference block temperature of a temperature measuringsystem similar to that depicted by Fig. V-II. If each time a test data point is recorded,multiple scan recordings are made for each of the thermocouples ~nd if a multiple scanaverage (Xij) is calculated for each thermocouple, then the average (Xj) for all recordings ofthe jth thermocouple is

X.J

Klx ..i=l IJ

K(V40)

where K is the number of multiple scan recordings for the jth thermocouple.

The grand average (X) is computed for all monitor thermocouples as

x

Nl Xjj=l

N

(V41)

The precision index (Sx) for the data acquisition and reduction processes is then

sx

N K _ 2l l (x .. - x.)j=l i=l IJ J

Nl (K - 1)j=l

(V42)

93

The degrees of freedom associated with Sx are

Ndfx = ~ (K - 1)

i=l(V-43)

Bias limits for the data acquisition process will be estimated by the instrumentation engineer.

Error sources accounted for by this method are:

1. Ice point reference precision error

2. Reference block temperature precision error

3. Recording system resolution error

4. Recording system electrical noise

5. Analog-to-digital conversion error

6. Chromel-Alumel thermocouple millivolt outputversus temperature curve-fit error

7. Computer resolution error

Several errors which are not included in the monitoring system statistiCs are

1. Conduction error (Be)

2. Radiation error (BR )

3. Recovery error (By)

4. Calibration error (Bl)

These errors are a function of probe design and environmental conditions. Detailedtreatment of these error sources is beyond the scope of this work~ Several goodreferences which should provide the background required to complete an error analysisare listed below:

1. "A Design Procedure for Thermocouple Probes," Laurence B. Haig, GeneralMotor Corp., SAE Preprint l58C, SAE National Aeronautics Meeting, April1960.

2. "Experimental Determination of Time Constants and Nusselt Numbers forBare-Wire Thermocouples in High-Velocity Air Streams and Analytic Approximation of Conduction .and Radiation Errors," Marvin D. Scadron and IsidoreWarshawsky, NACA TN 2599, January 1952.

3. "Recovery Corrections for Butt-Welded, Straight-Wire Thermocouples inHigh-Velocity, High-Temperature Gas Streams," Fredrick S. Simmons, NACARM E54G22a, September 1954.

94

AEDC-TR-73-5

4. "Radiation and Recovery Corrections and Time Constants of SeveralChromel-Alumel Thermocouple Probes in High-Temperature, High-Velocity GasStreams," George E. Glawe, Fredrick S. Simmons, and Truman M. Stickner,NACA TN 3766, October 1956.

5. "Performance of Three High-Recovery-Factor Thermocouple Probes forRoom-Temperature Operation," Marvin D. Scadron, Clarence C. Gettelman, andGeorge J. Pock, NACA RM E50129, December 1950.

6. "Recovery Characteristics of a Single-Shielded Self-Aspirating ThermocoupleProbe at Low Pressure Levels and Subsonic Speeds," C. E. Willbanks,AEDC-TR-7l-68, April 1971.

5.3.2.2 Resistance Thermometers

One tried and proved procedure for evaluating the data acquisition and reductionerrors associated with resistance thermometer temperature measurements is as follows:

1. Wirewound resistors, matched to ±0.01 percent of the desired value, aresubstituted in place of resistance thermometers.

2. With the exception of the substitution resistors, the temperaturemeasurement system should be identical to that used during engine testing.Components of the system (Fig. V-12) are

a. Bridge completion network (BCN)

b. Power supply

c. Recording system electrical calibration

d. Switching components

Bridge CompletionNetwork (BCN)

Power Supply

e. Signal conditioning equipment

f. Analog-ta-digital converter

g. Magnetic tape recorder.

Analog-To-DigitalConverter

SignalConditioningEquipment

Fig. V-12 Temperature Data Recording Calibration

95

AEDC-TR-73-5

3. Select at least four resistance thermometer channels.

4. Follow normal pre-run procedures for setting up the recording system.

5. Make several (ten or more) multiple scan (ten or more) recordings.

6. Reduce the data by means of the normally used gas turbine engine datareduction program a~d calculate a multiple scan average (Xij) for eachrecording of each channel. The average (Xj ) for all recordings on the jthchannel is

X.J

M.J

L x..i= 1 1J

M.J

(V-44)

where Ml_ = the number of recordings on the jth channel. The grandaverage (X) is computed for all channels:

NL X.j=l J

N

where N = number of channels tested.

(V-45)

The precision index (Sx) for the data acquisition and reduction processes is then

sx

N M.J _ 2

L L (x .. - x.)j= 1 i=l 1J J

NL (M. - 1)j=l J

(V-46)

1he degrees of freedom associated with the precision index are

Ndfx = L (M.-l)

j=l J

Bias limit estimates are left to the judgment of the instrumentation engineer.

(V-47)

The only error sources nG.. accounted for are bridge completion network (BCN)environmental effects and errors resulting from the dynamics of an engine test. The firstwill be accounted for if the BCN environment is at engine test conditions. If not, thenlaboratory tests are required to determine the effects of BCN temperature variations ontemperature measurements. Errors resulting from the dynamics of an engine test areaccounted for by means of multiple measurement statistics. From multiple measurements,the pooled within-run precision error (Sw r) and the pooled run-to-run precision error(Srr) for each resistance thermometer may be obtained. Derivations and formulation for

96

AEDC-TR-73-5

calculating SWI and SrI are in Appendix C. If thermometer and BCN are not calibrated asa unit (interchangeable BCN's) then a third multiple measurement statistic (See) must becalculated. This statistic is referred to as the pooled calibration-to-calibration precisionerror. Derivations and formulation for this statistic are also in Appendix C.

5.3.3 Temperature Measurement Error Summary

5.3.3.1 Thermocouples

The precision index for temperature measurements made with thermocouples is

where

ST = ±.... I S 2 + S- 2 + S 2 + S 2-" 1 X wr rr

Sl = calibration hierarchy precision index

Sx = data acquisition and reduction precision index

Sw I =within-run precision index (multiple measurements only)

Srr =run-to-run precision index (multiple measurements only)

(V48)

The degrees of freedom associated with ST are

(5 12

+ S_2 + S 2 + 2)2SdfT

X wr rr (V49)S 4 S 4 S 4 S 4

1 2 wr rrdf + df2

+ -- + --df df1 wr rr

Bias limits for the measurements are

where Bl = calibration hierarchy bias limits

Ex = data acquisition and reduction bias limits

Be = conduction error bias limits

BR = radiation error bias limits

By =recovery factor bias limits

(V-50)

5.3.3.2 Resistance Thermometers

The precision index for temperature measurements made with resistancethermometers is

(V-51 )

97

AEDC-TR-73-5


SX = data acquisition and reduction precision index

Sw r = within-run precision index (multiple measurements only)

Srr = run-to-run precision index (multiple measurements only)

See = calibration-to-calibration precision index (multiplemeasurements only)

Se = BCN environmental effects not included in Sx

The degrees of freedom df associated with this precision index are

(5 12 + 5- 2 + 5 2 + 5 2 + 5 2 + s e2)2

dfTX wr rr cc

(V-52)5 4 5- 4 4 4 4 45 5 5 s1 X wr rr cc e

F+ dfX+ -- + -- + -- + dfdf df df1 wr rr cc e

Bias in the temperature measurement is estimated by

where be is the bias resulting from BCN environmental effects.

(V-53)

If special tests are not performed to determine the overall effects of data acquisitionand reduction processes, each of the elemental errors must be evaluated as in Section III.The precision index for the temperature measurement is then

(V-54)

where S1, S2, and S3 are the root-sum-square of the elemental precision errors in thecalibration hierarchy, data acquisition process, and data reduction process, respectively.

The degrees of freedom for the temperature measurement are

(5 12 + 5 2 + 5 2 + 5 2 + 5 2 + 5 2 + s e2) 2

dfT2 3 wr rr cc

5 4 5 4 5 4 S 4 S 4 S· 4 s 4 (V-55)1 2 3 wr rr cc e

df1

+ df + df + + - + -- + -df df df df

2 3 wr rr cc e

Bias limits for the temperature measurement are calculated by root-sum-squaring theelemental biases.

±~ f bi2

98

(V-56)

AEDC-TR-73-5

SECTION VIAIRFLOW

6.1 GENERAL

Airflow rate measurements in gas turbine engine systems are generally made with oneof three types of flowmeters: venturis, nozzles, and orifices. Selection of the specific type offlowmeter to use for a given application is contingent upon a trade-off betweenmeasurement accuracy7 of requirements, allowable pressure drop, and fabricationcomplexityIcost.

Flowmeters may be further classified into two categories: subsonic flow and criticalflow. With a critical flowmeter, in which sonic velocity is maintained in the flowmeter, massflow rate is a function only of the upstream gas properties. With a subsonic flowmeter,where the throat Mach number is less than sonic, mass flow rate is a function of bothupstream and downstream gas properties.

Equations for the indicated mass flow rate through nozzles, venturis, and orifices arederived from the continuity equation;

where

W = pAY

W= mass flow rate, lb/sec

p = density of the gas at the meter throat, lb Ift3

A =cross-sectional area of the throat, ft2

V =gas velocity at the throat, ft/sec

(VI-I)

In using the continuity equation as a basis for indicated flow equation derivations, it isnormal practice to assume conservation of mass and energy and one-dimensional isentropicflow. Expressions for indicated flow will not yield the actual flow since actual conditionsalways deviate from ideal. An empirically determined correction factor, the dischargecoefficient (Cd), is used to adjust indicated to actual flow:

wC

actuald =

Windicated(VI-2)

Gas flow through venturis and nozzles closely follows the contour of the constriction,which results in geometric control of the flow area; in an orifice, the flow- stream iscontracted downstream of the constriction at the vena contracta. Location and size of thevena contracta (minimum flow area) vary with flow rate. As a result; flow through venturisand nozzles can be better described analytically, and therefore, flow measurements madewith venturis and nozzles are potentially more accurate than flow measurements made withorifices.

7For a definition of terms used in this Handbook, see the Glossary in Section IX.

99

AEDC-TR-73-5

Since, in the evaluation of turbine engine performance, engine inlet total airflow is oneof the three most important measurement parameters, the requirement for highmeasurement accuracy generally outweighs all other considerations. Hence, it isrecommended that a venturi system, operating at critical flow conditions, be used.

In the sections to follow, airflow measurement techniques will be discussed on thebasis of the two basic systems: subsonic flowmeters and critical flowmeters.

6.2 AIRFLOW RATE MEASUREMENT TECHNIQUES

6.2.1 Subsonic Flowmeters

6.2.1.1 Venturis and Nozzles

Figure VI-l schematically depicts a representative venturi and nozzle design.i-~>!,

I"~ VJ>

Fig. VI-1 Schematic of TypicalVenturi and Nozzlewith MeasuringStations

2

Ic::--- ---Flow

Flow ~-_................. ------~!O_-, ,--,--

I Venturi

Measuring Station 1

I

Nozzle

Flow rates through subsonic venturis and nozzles may be calculated by means of Eq.(VI-3) which is derived from the continuity equation.

(VI-3)

where W=airflow rate, Ibm/sec

d = meter throat diameter, in.

Cd = meter discharge coefficient (unitless)

Fa = meter thermal expansion correction factor(unitless) (evaluated at meter skin temperature)

PI = upstream total pressure, Ibf/in.2

TI = upstream total temperature, oR

P2 = throat static pressure, Ibf/in.2

K =ratio of specific heats at T1 (unitless)

M =molecular weight of the flowing gas, Ibm/Ibm-mol

Z = compressibility factor (unitless)

R =universal gas constant, 1545 ft Ibf/lbm-mol-o R

g = dimensional constant, 32.1741bm-ft/lbf-sec2

100

AEDC-TR-73-5

Equation (VI-3) is a general expression for the case where meter upstream totalpressure and temperature are measured. A typical example of this type of measurementsystem is the airflow measurement at the compressor inlet station of a gas turbine engine inthe flow channel (nozzle) formed by the engine inlet ducting and the engine inletcenterbody or spinner.

It should be noted that derivation of Eq. (VI-3) from the continuity Eq. (VI-I)requires the assumption of a calorically perfect gas, in which case Z = 1. It i~ furtherassumed that the upstream total temperature (T1) has been corrected for conduction,convection and radiation losses and for probe recovery factor, as specified in Section5.3.2.1. For real gases, values of the specific heat ratio (K), the molecular weight (M), andthe compressibility factor (Z) are a function of the gas composition, pressure, andtemperature. The equation is suitable for gases normally encountered in turbine enginework; the value of K should be evaluated at the upstream stagnation conditions, whereas thevalue of Z should be evaluated at the meter throat static conditions. The meter dischargecoefficient (Cd) is a function of the meter construction, the measurement and calculationtechniques, and the gas properties at the meter throat station.

Taylor's series expansion of Eq. (VI-3) results in Eq. (VI-4) for calculation" of theprecision index (Sw) and Eq. (VI-5) for the bias limit (Bw). Errors associated with values of1r, Fa' K, g, IJ., Z, and R in Eq. (VI-3) are assumed negligible in the following calculations.

K-I (K~~~)-i(:"1r,I

(K-I~' )- K(-,-(VI-4)IJeY(lr 1 K PI PI

SP22

CdSC d + d Sd + ,~I ST I + KPI+ f K-'] SPI!+KP2 - [ K-I

~2) K 21_(~)K21- ~

I I

(K~~~f(:'1 (K-I~') K(-,-1 JCY(1 r B 2+ 1 K PI PI 2

Cd Bc d + d Bd + ;T I BTl + ~;', + f K_lj PI iP;- [ K-I Bp2

~2) K 21_(~)K (VI-5)21- ~

Note that each term in Eqs. (VI-4) and (VI-5) has been divided by W to simplify theequations. The following list of partial derivatives is given as an aid for the analyst:

awad 2W - d

101

AEDC-TR-73-5

awaC d 1-W- = Cd

awaT I .--=w

1

- 2T I

aw !!.. d2 C F ~gM-a = 4 d a ri' Z R

P2

2

~K K-I .

K K- 1 12K P2

~ 1- (P2\K-I PI)

awap 2 1

W - Kp~

awap

2

!!.. d2 C F -[2gM4 d a"~

~ 1- (P 2\

K-I PI)

awaP I K-1

--W='KPI

102

AEDC-TR-73~5

In Eqs. (VI-4) and (VI-S), it is assumed that PI and P2 are independent measurements ontwo different pressure transducers. If PI is measured and a ~P equal to (PI - P2) ismeasured, Eqs. (VI-4) and (VI-S) reduce to

±W (aw/apI )2 {~w/aI1P S )2

+ W SP I + \ w I1p

(VI-6)

(VI-7)

where

aw/ap I

W(VI-8)

aw/al1pw

-1K-I

K

(VI-9)

Several recommended practices are available for designs of subsonic venturis andflow nozzles. In this document, the ASME recommended practice contained in "FluidMeters, Their Theory and Applications," (6th Edition, 1971) will be used. When using aspecific design practice, extreme care must be exercised to ensure that not only the meterdesign, but also the measuring systems, meet the~ recommended practice, since thetabulated values of meter constants apply only to a specific combination of meter designplus measurement technique.

The adiabatic isentropic equation for flow of an ideal-compressible flu,id through aventuri or flow nozzle is (from the cited ASME reference)

2

(VI-l 0)

103

AEDC·TR·73·5

where Wt = theoretical weight rate of flow

a = meter throat area

Pi = upstream static pressure

P2 = downstream static pressure

71 = specific weight of gas at station I

r '= pressure ratio, P2/pl

k = ratio of specific heats, cp Icv

~ =ratio of throat diameter to pipe diameter

g =dimensional constant

(VI-II)

For simplification, an adiabatic expansion factor (Ya) is defined as

1

Ya = J.(~~·_r k~1)~1~~4_)2k-l) l-r 2

, k1-[34 r

Values of Ya are graphically presented in the ASME reference. Substituting Eq. (VI-II) intoEq. (VI-lO) results in

(VI-l 2)

Applying the meter discharge coefficient (C) to correct indicated flow to actual flowand the area factor (Fa) to account for thermal expansion of the meter throat area andrearranging the terms of Eq. (VI-12) result in the following expression:

(VI-l 3)

where d = meter throat diameter.

Values of venturi and flow nozzle discharge coefficient are graphically presented in theASME reference.

Expres'sing the gas specific weight ('Y) in terms of measurable parameters of pressureand temperature, accounting for gas composition, and utilizing the gas compressibilityfactor (Z) to account for deviations of real gases from an ideal gas give

Yl

104

(VI-14)

where PI = upstream static pressure

M = gas molecular weight

Z = gas compressibility factor

R =universal gas constant

TI = gas temperature, corrected for lossesand probe recovery factor

AEDC·TR·73·5

Substituting into Eq. (VI-13) and utilizing nomenclature consistent with that used inEq. (VI-3) result in

6.2.1.2 Orifices

(VI-IS)

--------- ---

Measuring Station 1T

2T

As with venturis and nozzles, severalrecommended practices exist for orificedesigns. A typical orifice, with measuringstations, is shown schematically in Fig.VI-2.

Flow - o

----~..........._--Fig. VI-2 Schematic of Typical Orifice with

The ASME recommended practive for Measuring Stationsthin-plate orifice design contained in"Fluid Meters, Their Theory and Applications," (6th Edition, 1971) will be utilized in thissection. Care must again be exercised to ensure that not only the orifice design requirementsare met, but also the recommended practice for pressure and temperature measurements arefollowed, since, for thin-plate, square-edged orifices, values for both the meter expansionfactor and the discharge coefficient have been empirically determined. for the specifiedmeter design and pressure measurement system.

Flow through an orifice may be calculated by use of Eq. (VI-IS), using appropriatevalues of discharge coefficient (Cd) and expansion factor (Y). When metering gases withventuri tubes or flow nozzles, the expansion which accompanies the change in pressure takesplace in an axial direction only, because of the confining walls of these meters, and theadiabatic expansion factor (Ya) compensates for this unidirectional expansion. With athin-plate orifice, .there are no confining walls, and the expansion takes place both radiallyand axially. To account for this multidirectional expansion, an empirical expansion factor(YI ) is used. Values of YI are graphically presented in the ASME reference. For all thereferenced pressure tap locations except pipe taps, the empirical expression for the orificeexpansion factor is

(VI-I 6)

105

AE D CoT R-73-5

where {3 =ratio of orifice diameter to pipe diameter

Pl =upstream static pressure

P2 =downstream static pressure

k =ratio of gas specific heats

Error equations for orifices when upstream static pressure is measured on anabsolute transducer and ~P is measured on a differential transducer are as follows (Errorsassociated with values of 1T, (3, Ya, Fa, g, /.1, Z, and R in Eq; (VI-IS) are assumednegligible in the following calculations.):

~2Sd) 2 (Sc d)2 (-ST1)2SW = W d + C + 2T

d 1(3S~~ 2

+ 2~p) (

_S ) 2

+ 2:: (VI-17)

(VI-18)

The Welch-Satterthwaite formula must be used to detennine overall degrees of freedomassociated with Sw. If the degrees of freedom are greater than 30 for each term in the flowelTor equation, the overall df will be greater than 30, and t95 = 2 is used in thedetermination of flow uncertainty:

Critical Venturi

Figure VI-3 schematically depicts a representative critical venturi flowmeter installed inthe inlet ducting upstream of a turbine engine. Measuring station designations used in thissection represent AEDC standard practice.

Measurement· ooStationT

Flow

Venturi

Fig. Venturi Flowmeter InstallationEngine

When a venturi flowmeter is operated at critical pressure ratios, i.e., Pl N IPoo is a, minimum, the flow rate through the venturi is a function of the upstream conditions only

and may be calculated from

106

where

7T d2 PW C F C* __0_0

IN = -4- d a --

yT 00

WIN =airflow rate through venturi, Ibm/sec

d =venturi throat diameter, in.

Cd =meter discharge coefficient

Fa =meter thermal expansion correction factor

Poo =upstream total pressure, Ibf/in. 2

Too =upstream total temperature, oR

C* =critical flow factor,

AEDC-TR-73-5

(VI-l 9)

=

1

~K+l ~2

2 K-l K gM Ibm-yOJ1

K+~ QZR)' lbl-sec(VI-20)

K =ratio of specific heats evaluated at To 0 (unitless)

g =dimensional constant, 32.174 lbm-ft/lbf-sec2

R =universal gas constant, 1545 ft-lbf/lbm-mol 0 R

Z =compressibility factor (unitless)

M =molecular weight of flowing gas, Ibm/Ibm-mol

The indicated upstream total temperature measurement must be corrected for thermallosses and probe recovery factor as discussed in Section 5.3.2.1.

For a venturi designed according to the specifications set forth in ASME Paper No.6l-WA-2ll, the value of Cd may be theoretically determined as described in the referencedpaper.

Equations for determination of precision index (S) and bias limit (B) are as follows(Errors associated with values of 1r, Fa, and C* in Eq. (VI-19) are assumed negligible inthe following calculations.):

+ .(~ S )2 + (.2- S )2 (VI-2l)2T T p P

00 00 00 00

and

(VI-22)

Again the degrees of freedom will be determined by the Welch-Satterthwaite formula.If the degrees of freedom are greater than 30 for each error term, then t95 can be assumedequal to 2.

107

AEDC-TR-73-5

The reader is cautioned about two pitfalls common to gas flow measurement withcritical flowmeters.

1. Mass flow rate is a function of upstream total pressure. Therefore, approachvelocity corrections must be made if wall static pressure measurements areused. As a rule, if the velocity is maintained at 50 ft/sec or less, use of theisentropic flow relations to obtain total pressure from the measurement ofwall static pressure will result in negligible errors in flow rate determination.

2. It is essential that the choked condition be maintained if Eq. (VI-l 9) is usedto calculate airflow. A critical flowmeter can, however, be operatedsubcritically; in this event, the flow rate calculation method is identical tothat shown by Eq. (VI-3) using the venturi throat static pressure (Pl N ) (Fig.VI-3) for the P2 term of Eq. (VI-3). However, the uncertainty of the airflowmeasurement will be slightly greater than for the case of critical operation ofthe venturi.

6.2.3 Calibration Techniques

Venturi, nozzle, and orifice calibrations are performed to determine the dischargecoefficient.

6.2.3.1 Calibration by Calculation

Discharge coefficients for critical flow venturis and nozzles can be calculated with anaccuracy greater than that for the experimentally determined Cd provided that the designcriteria and considerations discussed in Section 6.3 .1.1 are adhered to. In many instances,determination of Cd by calculation is so accurate that critical venturis are consideredprimary standards when associated pressure and temperature measurements are traceable tothe NBS.

6.2.3.2 Experimental Calibration

Flowmeters for which the discharge coefficient cannot be accurately calculated may becalibrated experimentally by one of three ways:

1. Flow in series with a critical venturi for which the Cd can be accuratelycalculated.

2. Flow a known volume of liquid through the meter at fixed temperature andpressure and appeal to dynamic similarity to translate the results tocompressible flow.

3. Traverse the meter with a pitot static probe to define the flow profile. Theresults of a complete traverse, including boundary layer measurements,constitute a flow profile which will yield total flow when integrationbeneath the profile is performed. Definition of the relationship betweentotal flow and measured pressures and temperature constitutes a calibration.

108

AE DC·TR·73·5

6.2.3.3 Calibration by Fabrication

The ASME has determined the discharge coefficients for a variety of flowmetersincluding venturis , nozzles, and orifices. The ASME discharge coefficients have beenpublished along with Cd error tolerances in the ASME publication "Fluid Meters, TheirTheory and Applications," (6th Edition, 1971). Care must be taken to ensure that meterdesign and measurement systems conform to those specified by the ASME.

If a flowmeter does not comply with ASME design tolerances, ASME dischargecoefficient error tolerances do not apply, and calibration is required for flow measurementuncertainty analysis.

6.3 ELEMENTAL ERROR SOURCES

Number ofDeterminations

0.985 0.995 1.00

Discharge Coefficient, Cd

Fig. VI-4 Discharge Coefficient ErrorDistribution

6.3.1 Discharge Coefficient

The values of Cd may exceed onebecause of the combination of measurement error and calculation technique andthe assumptions utilized in determiningthe value of discharge coefficient. The best value of Cd to use for data reduction is:

Discharge coefficient (Cd) errors ofconcern are primarily biases, and the signis unknown, i.e., the error is just as likelyto be in one direction as the other. Theerror distribution will probably be nearnormal (Fig. VI-4).

N~ Ci=l d i

N

(VI-23)

and

±

(VI-24)

6.3.1.1 Calculated Cd

Discharge coefficients for critical flow venturis may be calculated provided the venturidesign conforms to criteria set forth in "A Theoretical Method of Determining DischargeCoefficients for Venturis Operating at Critical Flow Conditions," by Robert E. Smith, Jr.,and Roy J. Matz, Journal of Basic Engineering, December 1962, page 434, ASME Paper No.6l-WA-211. Venturi design considerations for inlet contour, inlet to throat area ratio, andthroat Reynolds number are given in the above paper. Errors of less than ±O.! percent will

109

AE DC·TR~73-5

result from assumptions made in the calculation. This error is considered an unknown biasand is relatively small as compared with errors common to other sources.

6.3.1.2 Experimentally Determined Cd

Critical flowmeters which do not fulfill the conditions set forth in Section 6.3.1.1 mustbe calibrated experimentally to ensure maximum accuracy in Cd determination.

Three common methods of experimental calibration are-:

1. Comparison with a critical flow standard.

2. Calibration by traversing the flowmeter.

3. liquid calibration.

6.3.1.2.1 Comparison with a Standard Flowmeter

Venturis , nozzles, and orifices, sonic or subsonic, may be calibrated in series with acritical flow venturi of known Cd.

For the purpose of illustration, assume two choked venturis in series (Fig. VI-S) andfurther assume that the inlet flow to both venturis satisfies the criteria set forth in theASME paper referenced in Section 6.3.1.1.

Plenum Plenum

Fig. VI-5 Calibration by Comparison

Assume that the Cd for venturi A has been calculated with a bias no greater than ±O.lpercent.

The mass flow rate through venturi A is

The mass flow rate through venturi B is

(fTd 2) ~ P 00 )w = _B_ e F e* B

B 4 dB aB y-T--oOB

110

(VI-2S)

(VI-26)

In series, WA = WB; therefore, the two equations may be set equal to each other andsolved for the product dB 2 Cd B' Simplification results from the assumption that the criticalflow factor (C*) is equal within the range of conditions to be encountered for each venturi.However, if dB is much greater than dA, significant real gas effects may be experienced inventuri A. Monitoring the temperature of each meter will allow thermal expansioncorrections, and

(VI-27)

Expanding Eq. (VI-27) in a Taylor's series results in Eqs. (VI-28) and (VI-29) for theprecision index and bias limit, respectively. Errors associated with values of FaA and FaBare assumed negligible in the following calculations:

The following example will indicate theeffect of typical errors ondischarge coefficientdetermination by comparison. Table XV liststhe elemental errors assumed for this· example.

Substituting thesevalues into Eq. (VI-28)for precision index andEq. (VI-29) for bias limitgives

Table XV Elemental Errors for Calibration by Comparison

Error Precision Error, Bias Error, Degrees ofSource Nominal percent percent Freedom

CdA 0.995 0 ±0.1 > 30

dA 10.0 in. 10.01 ±0.01

POOA 100. psi ±0.1 . 10.15

POOB 100. psi ±0.1 10.15

TOOA 600. OR ±0.16 10.08

TOOB 600. OR ±0.16 ,,'10.08

±0.18 percent

±O.24 percent

111

AEDC-TR-73-5

Since each of the elemental degrees of freedom is greater than 30, the overall degreesof freedom are also greater than 30; t95 =2.0, and

U ±[0.24 + 2(0.18)]

±0.60 percent

Obviously, determination of discharge coefficients by this method is less accurate thanthe same Cd if evaluated by calculation.

Note that the error in discharge coefficient evaluation is but one of several elementalsources of error in the calculation of airflow.

6.3.1.2.2 Calibration by Traverse

t------ d----~

Fig. VI-6 Flowmeter Throat Traverse

(pV)e

and

277 rWa = f J (p V)e rdr de

o 0

Flowmeter discharge coefficients mayalso be determined by traversing the meterthroat with a pitot static probe to define themeter throat mass-velocity profile. It isrecommended to traverse several meter throatdiameters, equally spaced circumferentially,to better define the mass-velocity profile. Theresult of a typical four-diameter traverse is anarray of gas stream measurements as shown inFig. VI-6. For convenience the measurementpoints on each diameter should be located atthe centers of equal area annuli.

For the general case of airflow measurement, the local mass-velocity is a function ofboth radial and circumferential position:

(VI-3D)

(VI-31 )

where Wa =total mass flow rate through meter throat

{pV)Q =local value of mass-velocity

r =radial location of local measurement

() =circumferential location of local measurement

The (pV)Q term may be obtained from the traverse data by solving Eq. (VI-3) for flowrate per unit area and using the local values of throat total and static pressure and totaltemperature:

112

AEDC~TR~73-5

(VI-32)

r--~x x

t t t t t tCIl ('I) "<:I' It) co ~

M M M M M M

0 Throat Area A

Fig. VI-7 Flat Mass-Velocity Profile

(pV)e = (W) = (p~\ ('p')i (~\ ~ _(E) K~J 2gMA e ',IT) f P e K-i) L ~ e ZR

The effect of a variation of (p V)Qwith circumferential position in thenozzle throat may be approximated byaveraging the local mass-velocity measurements at particular radial positions (r)common to each diameter traversed. Anaverage mass-velocity profile can then bedeveloped as shown in Fig. VI-7. Careshould be exercised to ensure thatboundary layer profiles are sufficientlydefined.

Integration beneath the profile defined by the average mass-velocities provides ameasure of total mass flow rate:

A-Wt = f (pV) r dA

o(VI-33)

where8~N~pV)~

(-) 8=1 L ~e ,rpV r = N (VI-34)

N = number of equally spaced circumferentialmeasurements taken at each radial position

dA = incremental flow area over which each averagemass-velocity (pV)r is effective

AThroat Areao

tWhether ~he profile is distorted or

undistorted, a valid approximation to thetotal flow given by Eq. (VI-33) may beobtained by summing the flows calculatedfor each incremental ring area (AR) asshown below:

Mass-velocity profile errors are dominated by two sources: profile distortion andinstrumentation errors. If the profile is distorted (Fig. VI-8) the cause of distortion shouldbe diagnosed and corrected, if possible. With a distorted profile, a larger number of diametertraverses and a greater number of pointsalong each diameter are recommended toprecisely define the existing profile.

R=i R=i

Wt = ~ WR = ~ (p V) r ARR=1 R=1 (VI-35)

Fig. VI-8 Distorted Mass-Velocity Profile

113

where Wt = total mass flow rate through meter throat

WR =mass flow rate through each incremental ring

(pV)r =average mass-velocity in each incremental ring

AR = area of each incremental ring

i =number of incremental rings

The precision index for each incremental flow is evaluated by expansion of Eq. (VI-32)in the Taylor's series:

(VI-36)

(VI-37)

Fig VI-9 Shaded Area Calculated asa Function of d1 and d2

The precision index S(p V)r may be obtainedas previously shown in Eq. (VI4) with elimination of the meter throat area terms (Sc d and Sd).

The precision index SA R is dependent onthe repeatability of the traverse probe radialposition indication dl and d2. Figure VI-9illustrates this dependence.

rrd 2 ri'd 2 (d 2 _ d 2)A=_1 2_=rr 1 2444

Taking the partials with respect to the probe radial locations gives

aA rrd land

aA -rrd2ad

l= 2 ad

2= -2-

2 2

~~~:' Sd,y (-rrd yCIA S ) (aA . )SA + ad2

Sd2

+ ~ Sd2

(VI-38)R CAl d l

where Sd 1 and Sd 2 are the precision indices for the respective probe radial locations.

The precision index for the total mass flow rate determination is then

(VI-39)

The airflow bias limit BwR cannot be calculated directly from the traverse data. Thiserror is due entirely to elemental bias errors imposed by the instrumentation and may 'beestimated by the use of Eq. (VI-S) and dropping the terms for diameter (d) and dischargecoefficient (Cd).

114

AEDC-TR-73-5

Solving Eq. (VI-19) for Cd in the case of a critical venturi results in

(VI-40)

Expanding Eq. (VI-40) in the Taylor's series results in Eq. (VI-41) and (VI-42) for theprecision index and bias limit, respectively. Errors associateq with values of 1T, Fa, and C*are assumed negligible in the following calculations:

ac ) 2 (ac )2aw

dSw. + ad d Sd

IN IN ~ac )2

+ aT d ST00 00

(ac \2

+ oP :0 Sp 00) (VI-41)

(ac )2 (ac )2 (ac )2+ ~ Bd + ar:- BToo + ~ Bp 00 (VI-42)

c 2d

where aC d aC d aC d aC daW 1N 1 aI>" 1 aT 1 ad 200 00c;- = WIN;~ -p;-c-' 2T Cd - -d

00 d 00

The degrees of freedom (df) associated with Sc d will probably be greater than thirty sothat t95 will equal 2.0. If this is not obvious then df must be calculated using theWelch-Satterthwaite formula.

Equation (VI-3) may be solved in similar fashion for the Cd of subsonic meters.Expansion in the Taylor's series will result in equations for the precision index and biaslimits for subsonic flowmeters.

6.3.1.2.3 Calibration by Liquid

Venturi, nozzle, and orifice discharge coefficients may be evaluated by liquidcalibration. Methods used are fully analogous to those detailed in Section IV. For thisreason, the treatment of errors involved with this calibration technique will not be discussedhere. However, the reader is warned that,. when this method of calibration is employed,published data on expansion factors must be used with the associated errors which will bereflected in the discharge coefficient. Further, an appeal to dynamic similarity through theReynolds analogy is required, and quantitative assessment of the uncertainty of thesimilarity is unavoidable.

115

AEDC~TR~73~5

6.3.1.2.4 Calibration by Fabrication

The ASME has cataloged discharge coefficients for a variety of venturis, nozzles, andorifices. The cataloged values are the result of an extremely large number of actualcalibrations over a period of many years. The results of this experimental work isdocumented in the ASME publication, "Fluid Meters, Their Theory and Applications," (6thEdition, 1971). Discharge coefficients cataloged in this ASME reference are applicable to allflowmeters which conform to this specification. Detailed engineering comparisons must beexercised to ensure that the flowmeter conforms to one of the groups tested before usingthe tabulated values for discharge coefficients and error tolerances.

A later ASME publication, "A Statistical Approach to the Prediction of DischargeCoefficients for Concentric Orifice Plates," by R. B. Dowdell and Yu-Lin Chen, Paper No.6~-WA/FM-6, may be useful in determining which Cd to use and in evaluating errorsassociated with it.

When an independent flowmeter is used to determine flow rates during a calibrationfor Cd, dimensional errors are effectively calibrated out. However, when Cd is calculated ortaken from the ASME reference, errors in the measurem~nt of pipe and throat diameterswill be reflected as bias errors in the flow measurement.

Dimensional errors in large venturis, nozzles, and orifices are generally negligible. Forexample, an error of 0.001 in. in the throat diameter of a S-in. nozzle will result in a 0.04percent bias in airflow.

6.3.2 Non~ldeal Gas Behavior and Variation in ~as Composit!ons

The equations given in preceding sections for calculation of gas flow rate (Eqs. (VI-3),(VI-IS), (VI-19)) are specifically valid for any calorically perfect gas. Non-ideal gas behaviorand changes in gas composition are accounted for by selection of the proper values forcompressibility factor (Z), molecular weight (M), and ratio of specific heats (K) for thespecific gas flow being measured.

For the specific case of airflow measurement, the main factor contributing to variationof composition is the moisture content of the air. Though small, the effect of a change in airdensity due to water vapor on airflow measurement should be evaluated in everymeasurement pr~cess.

6.3.3 Thermal Expansion Correction Factor

The thermal expansion correction factor (Fa) corrects for changes in throat area causedby changes in flowmeter temperature.

For steels, a 30°F flowmeter temperature difference, between the time of. a test andthe time of calibration, will introduce an airflow error of 0.06 percent if no correction ismade. If flowmeter skin- temperature is determined to within ±SoF and the correction factorapplied, the resulting error in airflow will be negligible.

116

AEDC-TR-73-5

6.3.4 Ratio of Specific Heats and Compressibility Factor

The ratio of specific heat (K) at constant pressure to specific heat at constant volume(cp/cv =K) and the compressibility factor (Z) are functions of gas stream composition,pressure, and temperature and may be obtained from gas tables. When used in the equationsprovided for calculation of flow rate, the value of K should be evaluated at upstreamstagnation conditions, whereas the value of Z must be evaluated at the throat staticconditions.

When values of K and Z are obtained_ at the stated pressure and temperatureconditions, airflow errors resulting from errors in K and Z will be negligible.

6.3.5' Measurement Systems,

Errors associated with pressure and temperature measurements and their respectiveprobe recovery factors are described in Section V.

6.4 PROPAGATION OF ERROR TO AIRFLOW

6.4.1 Critical-Flow Venturi

For an example of propagation of errors in airflow measurement using a critical-flowventuri, consider a venturi (designed according to criteria presented in ASME Paper No.61-WA-211) having a throat diameter of 21.81 in. operating with dry air at an upstreamtotal pressure of 12.78 psia and an upstream total temperature of 478.7°R. Equation(VI-19), repeated below, is the flow equation to be analyzed:

K+l

i[ 2 ) K - 1 (KigM)' P 00

\K+ 1 ,Zit, VT00

Assume, for this example, that the theoretical discharge coefficient (Cd) has beendetermined, using the procedures outlined in ASME Paper No. 6l-WA-211, to be 0.995.Further assume that the thermal expansion correction factor (Fa) and the compressibilityfactor (Z) are equal to 1.0. Table XVI lists nominal values, bias limits, precision indices, anddegrees of freedom for each error source in the above equation.

Note that, in the following table, airflow errors resulting from errors in Fa, Z, K, g, M,and R are considered negligible.

From Eq. (VI-19), airflow is calculated as

3. ~42 (21.81) 2 x. 0.995 x 1.0

115.5 Ibm/sec

2.401

'I~)0.401 (1.401 x 28.95 x 32.174_\ x 12.78

\2.401 \: 1545 I V478.7

117

AEDC·TR.73.5

Table XVI Airflow Measurement Error Source

Error PrecisionSource Nominal Value Bias Limit Index df Uncertainty

Poo 12.78 psia ±a.04 psia ±a.02 psia ' 15 ±a.08 psia

Too 478.7°R ±1.8°R ±O.20oR 100 ±2.2°R

d 21.81 in. ±a.001 in. ±O.OOl in. 100 ±a.003 in.

Cd 0.995 ±a.003 ... -_. ±a.003 in.

Fa 1.0 ... ..- -.- ...

Z 1.0 ..- ... ... ...

K 1.401 ... -_. ... ...

g 32.174 lbm·ft/ ... ._- ... ...Ibf·sec2

M 28.95 Ibm/Ibm·mo ... -.. ..- ...

R 1545 Ibf-ft/ ... ... -.. ...

Ibm·moI·oR

Equations for determination of precision index (8) and bias limit (B) were given byEqs. (VI-21) and (VI-22):

± ('0.02)2 (-0.20)2 ( 0 \2 (2XO.001)212.78 + 2 x 478.7 + 0.995) + 21.8 2

±0.0012 = ±0.12 percent

S = ±0.12 percent x 115.5 lbm/sec = ±O.139 lbm/secWIN

(_BT01

2(BCd~:L (2Bd)2

2T + C + i:"00 d ~(ai

± ( 0.04)2 (-1.8) 2 (0.003)2 (2 x 0.001)2.12:78 + 2 x 478.7 + 0:995 + 21.g~

± )(0.0024)2 + (-0.0018)2 + (0.003015)2 + (0.000004)2

±0.0046 = ±0.46 percent

±O.46 percent x 115.5 lbn!sec = ±O.531 lbm!sec

118

By using the Welch-Satterthwaite formula, Eq. (1-11), the degrees of freedom for thecombined precision index is determined from

S 4d

+dfd

S 4T

00

( S. P 2 + ST 2 + Sd2)2df = 00 00

Sp 400

which results in an overall degrees of freedom> 30, and therefore a value for tg 5 of 2.0.

Total airflow uncertainty is then,

±[0.0046 + 2(0.0012)]

±0.007 = ±0.7 percent

±O.7 percent x 115.5 Ibm/se c = ±O.81 Ibm/ sec

6.4.2 Subsonic Orifice

Consider a 2-in. orifice installed in a 4-in. pipe and used for airflow measurement. Themeter is a square-edged orifice with flange taps, and the value for Cd has been taken fromthe ASME references.

Equation (VI-15) is theflow equation to be analyzed.The variables to be consideredare d, Cd, ~p, PI, and TI .

Indicated total temperature(TI) should be corrected forlosses and probe recoveryfactor.

From the ASME references, Cd =0.6025 ± 0.58% (2sigma). Table XVII lists nominals, precision indices and biaslimits for each error source.

Table XV"II Airflow Error Source

Error .Precision Index

Nominal Value Bias LimitSource (S)

d 2.0 in. ±O.OOI in. ---Cd 0.6025 ±O.0017 ---L\, 5.00 psid ±O.005 psid ±O.OIO psid

PI 20.00 psia ±O.02 psia ±O.04 psia

TI 520.00 OR ±1.00 OR ±O.25 OR

(-0.04 )2

+ 2 x 20.0(-0. 25 ) 2 (3 x 0.0 1) 22(520) + 2 x 5.0

From Eqs. (VI-l 7) and (VI-l 8), with Sd and Sc d =0,

Sww=

0.0031714 0.32 percent

119

AEDC-TR-73-5

BWW -

~2 X 0.001)2 + (0.0017)2 (-1.0 )2 (3 X 0.005)2 (-0.02)2~ 2.0 0.6025 + 2 X 520 + 2 X 5.0 . + 2 X 20.0

= 0.0035194 = 0.35 percent

If the overall degrees of freedom are assumed to be greater than thirty, then

±(0.35 + 2 X 0.32)

120

±0.99 percent

AEDC-TR-73·5

SECTION VIINET THRUST AND NET THRUST SPECIFIC FUEL CONSUMPTION

7.1 GENERAL

This section details an error analysis for net thrust and net thrust specific fuelconsumption at an altitude test facility. In order to calculate net thrust, engine gross Get)thrust must first be determined. Two independent techniques for the determination ofgross thrust can be used: (l) external forces, or scale force method and (2) internal forces,or momentum balance method. Both techniques are utilized for most gas turbine engineperformance programs when the determination of net thrust and thrust specific fuelconsumption is a primary requirement. Performance determined by each method can becompared for agreement in order to improve the confidence in thrust data. Error analysisis presented only for the external forces or scale force method. The relationship betweengross thrust and net thrust will be shown in Section 7.3.

The measurements associated with the determination of net thrust and net thrustspecific fuel consumption include pressure, temperature, force, fuel flow, and airflowmeasurements. Error analysis of measurement systems have been presented in priorsections as follows: temperature and pressure measurements (Section V), forcemeasurement (Section III), fuel flow measurement (Section IV), and airflow measurement(Section VI).

7.2 GROSS THRUST MEASUREMENT TECHNIQUES

7.2.1 Scale Force Method

The engine assembly and engine support mount are installed on a thrust stand whichis flexure mounted on a model support cart. The engine inlet duct system contains azero-leakage, labyrinth-type air seal. Resultant axial forces are measured by a strain-gageload cell. This installation permits the defining of a control volume (Fig. VII-I) whichallows the calculation of gross Get) thrust (FG) from easily measurable parameters.8

The freebody diagram associated with the scale force method of thrustdetermination is shown in Fig. VII-I.

The derivation of gross thrust (FG) from Fig. VII-l is

wA v A

'£F x = 0 = : 1 + AlP 1 + F s - Po f 1 dAA

J

Rearranging and combining terms give

WJVJ WA1V 1-g-.-+AlPJ-Po)= g +A1(PI-Po)+Fs

8Por a defmition of terms used in this Handbook, see Glossary in Section IX.

121

force measuring transducer output (scale force), Ibf

engine exhaust nozzle exit gas flow rate~ Ibm/sec

engine exhaust nozzle exit gas flow velocity, ft/sec

engine exhaust nozzle exit 2area, in.

AEDC·TR·73·5

WAI engine inlet airflow rate, Ibm/sec

VI engine inlet airflow velocity, ft/sec

g dimensional constant, 32.174 Ibm-ft/lbf-sec2

Al engine inlet duct cross-sectional area (aD), in~

PI = engine inlet duct static pressure, Ibf/in~

FS

WJ

VJ

AJ

PJ engine exhaust nozzle exit static pressure, Ibf/in~

Po free-stream (ambient) static pressure, Ibf/in~

Fig. VII-' Freebody Diagram for External Forces (Scale Force)Method of Determining Engine Gross (Jet) Thrust

Engine gross (jet) thrust in pounds force is, by definition,

therefore,

7.2.2 Momentum Balance Method

(VII-I)

The momentum balance method of thrust determination utilizes the nozzle throattotal pressure and temperature profiles obtained with a traversing probe and amathematical flow field integration to determine the nozzle stream thrust.

Theoretical calculations utilized in the momentum balance method assume the fluidto be inviscid, thenna1ly perfect, and non-heat-conducting. The method of calculationconsists of direct numerical integration of the equations for continuity, momentum, and

122

.. gas flow rate at nozzle exit, Ibm/sec

= gas flow velocity at nozzle exit, 'ft/sec

nozzle exit 2area, in.

= nozzle exit static pressure, lbf/in~

nozzle wall static pressure, lbf/in~

gas flow rate at nozzle throat, Ibm/sec

gas flow velocity at nozzle throat, ft/sec

dimensional constant, 32.174 Ibm-ft/lbf-sec2

nozzle throat area, in~

nozzle throat static pressure, lbf/in~

AEDC-TR-73-5

energy to define the gas state at the nozzle throat. The gas properties at the nozzlethroat are then used to determine the exit momentum and pressure-area forces requiredto obtain the nozzle stream thrust.

The freebody diagram associated with the momentum balance method of thrustdetermination is shown in Fig. VII-2.

. TraversingProbe

WTHv

TH..

g

ATH

=

PTH

wJ

vJ

AJ

PJ

Pw

Po = free-stream (ambient) static pressure, lb£/in~

Fig. VII-2 Freebody Diagram for Internal Forces (Momentum Balance)Method of Determining Engine Gross (Jet) Thrust

The derivation of gross thrust from Fig. VII-2 is

ATH ATH AJA

JA

J

:£Fx 0 fWTH:TH/g dA + fPTH dA + fPW dA - fWJv

/

gdA - f PJdA

0 0 ATH 0 0

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AEDC-TR-73-5

Engine gross (jet) thrust in pounds force is, by definition,A

JA

J

JWJV Jig JA dA + PJ dA - AJP 0

therefore,

o o

(VII-2)o o o

7.3 PROPAGATION OF ERRORS TO NET. THRUST

When FG has been obtained by either the scale force or the momentum balancemethod, or both, identical equations are used to obtain net thrust. The external force(scale force) method for measuring net thrust is used as the example, and the derivationof this method is shown in Section 7.2.1. The. Taylor's series method (Appendix B) ofpropagating error to net thrust is used.

The relationship of net thrust (FN) to gross thrust (FG) is

F N = F G '- F R

where

(VII-3)

(VII-4)

and Vo is the aircraft free-stream veloeity in ft/sec, WA 1 is the engine inlet airflow rate inIbm/sec, and g is a dimensional constant. Combining Eqs-:-(VII-l), (VII-3), and (VII-4)then results in the following equation for engine net thrust:

FN =WA1-g- (V 1 - V0) + A1(p 1 - po) + Fs (VII-S)

where

VI 2K:~~1 ~ _~:)K~1 (VII-6)

Vo

2KgR T 1 ~ _(~o)K~I (VII-7)K-I

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PI =engine inlet duct total pressure, Ibf/in.2

K = ratio of specific heats at TI

R = gas constant for air at TI, ft-Ibf/lbm-o R

All other parameters are as designated in Fig. VII-I.

Net thrust in terms of pressure, temperature, area, airflow, a gas constant, a ratio ofspecific heats, a dimensional constant, and force measurement now becomes

1 - ~ll) K~l 1 _ (:0) K~JJAJ{PJ - Po) + Fs

(VII-8)

The propagation formulas for the bias limit and precision index are derived from Eq.(VII-8) for FN.

The bias limit propagation formula is the weighted root-sum-square of the bias limitsfor WA I, g, R, K, TI, PI, PI, Po, AI, and Fs:

In the same way, the precision index propagation formula is the weightedroot-surri-square of the precision indices of WA I, g, K, R, TI , PI, PI, Po, AI, and Fs:

Errors associated with values of g, K, and R are generally assumed negligible.

The uncertainty for net thrust (FN) is calculated using the uncertainty formula:

(VII-II)

The following list of partial derivatives is given as an aid to the analyst:

125

AEDC-TR-73-5

1

:~:1 = ~ t:~~T~2~1~1

WAl I KgR J2

g ~K-I)T~

-B2~J

rl~ B2~]

PI - Po

1.0

1 K-l ~

W (~)K (~)K((K-I)RTIj2Alp P 2gK B - Al

o 1 2

where

and

By using Eqs. (VII-9) and (VII-IO), the partial derivative equations above, and theexample values listed in Table XVIII, the propagation of error to net thrust can bedetermined.

In this example, values for the partial derivative terms above were approximatedthrough the basic net thrust equation. First, the thrust level is determined from themeasured values using performance Eq. (VII-8). Then each measured value in Eq. (VII-8)is changed, independently, by the amount of its precision index, and the resulting changein FN is obtained. This process, repeated for each measured value, provides goodapproximate numerical values for each term in Eq. (VII-I 0). For example to obtain theapproximate value for the term .[(oFN jaWAdSWAl] of Eq. (VII-tO), determine FN (Eq.(VII-8)) with the measured values. Then, determine the change in FN(LlFN) by changingairflow (WAI) by the amount of its precision index. The change in FN resulting from thechange in WA1 is approximately equal to the term [(aFN jaWAl )Sw AI]' The identicalprocess is repeated for bias limits to obtain numerical values for the terms in Eq. (VII-9).

126

AEDC.TR·73-5

Table XVIIITypical Measurement and Uncertainty Values Used in Net Thrust for

Supersonic Afterburning Turbofan Engine

Flight Condition: 30,000-ft Altitude, Mach Number 0.9, Military Power

Nominal Bias Precision Uncertainty, Degrees ofComponent Value Limit Index U Freedom**

Fs, Scale Force, lbf 4,388 7.90 3.95 15.80 105Ibm - ft

g, Dimensional Constant, lbf _sec2 32.174 * * * *

AI, Inlet Duct Area (OD), in.2 984 0.050 0.050 0.160 12

PI, Inlet Duct Static Pressure, psia 6.50 0.0065 0.0065 0.0202 15

Po' Free·Stream Static Pressure, psia 4.31 0.0099 0.0099 0.0310 15

R, Gas Constant for Air at TI,ft -lbf

53.329 * * * *Ibm - "RTI, Inlet Duct Total Temperature, oR 477 ±2.48 ±O.19 ±2.86 100

K, Ratio of Specific Heats at TI 1.4034 * * * *

PI, Inlet Duct Total Pressure, psia 7.43 ±O.0074 ±O.Olll ±O.0312 15

Wf, Total Fuel Flow, lbm/hr 4662 ±6.06 ±5.l3 ±16.32 35

*Bias, precision, and uncertainty considered as negligible in thrust uncertainty.**Degrees of freedom determined by the Welch-Satterthwaite formula (Eq. 1-11).

By using Eqs. (VII-6), (VII-7), and (VII-B) and the measurement values and theiruncertainty components listed in Table XVIII, the propagation of error to Vo , VI, FR,FN, and TSFC are shown in Table XIX. Wal of Table XIX was obtained from SectionVI.

Table XIXDerived Measurement Uncertainty Values

Bias Precision Degrees ofParameter Nominal Limit Index Uncertainty Freedom

V0, Free-Stream Velocity, ft/sec 908.4 ±3.049 ±2.1l9 ±7.414 26

Vb Inlet Duct Velocity, ft/sec 463.4 ±2.684 ±3.057 ±8.981 26

WaI , Inlet Duct Airflow, Ibm/sec 115.5 ±O.531 ±O.139 ±O.825 16

FR, Ram Drag, lbf 3261. ±18.563 ±8.559 ±35.681 38

FN, Net Thrust, lbf 4945 ±12.498 ±7.471 ±27.440 59

TSFC, Net Tluust Specific Fuel0.943 ±O.0027 ±O.0018 ±O.0063 94Consumption, lbm/lbf-lu .

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7.4 PROPAGATION ERROR TO THRUST' SPECIFIC FUEL CONSUMPTION

TSFC WF Ibm/hr.

FN'~(VII-12)

where WF is total engine fuel flow (lbm/hr), and FN is net thrust (lbf).

The error analysis for.,fuel flow is shown in Section IV, and the error analysis fornet thrust is shown in Section 7.3.

The bias limit propagation formula is the root-sum-square of the bias limits for WFand FN weighted by the partial derivatives (Appendix B):

B 2 _ +~aTSFC B)2 (a'JSFC BFN.)~TSFC - -~awF WF + \aF N J (VII-l 3)

In the same way, the preCISIon index is calculated as the root-sum-square of theprecision indices for WF and FN :

(VII-14)

The partial derivatives for the terms in Eqs. (VII-13) and (VII-14) are

aTSFC 1awF- F N

and

Equations (VII-13) and (VII-14) are now evaluated using the above partials and themeasurement values and uncertainty components from Tables XVIII and XIX for WF andFN, respectively.

± .)[(0.0002022)(6.06)]2 + [(0.0001906}(12.498)]2

±0.0027 lbm/hr/lbf

±/[(0.0002022)(S.13)] 2 + [(0.0001906)(7.471)] 2

± 0.0018 lbm/hr/lbf

The uncertainty for TSFC is calculated using the uncertainty formula:

U = ±(B + t 9 5S)

Here t95 is = 2.0 since the degrees ot freedom are greater than 30.

U ±[0.d027 + 2(0.0018)]

= ± 0.0063 lbmlhr/lbf

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SECTION VIIISPECIAL METHODS

8.1 GENERAL

This section treats several methods for speCial situations or conditions:

1. Measurement uncertainty9 for multi-engine installation (similar engines).

2. Measurement uncertainty for the single stand, single engine process incomparison with the many stand, many engine process.

3. Confidence interval for uncertainty when biases are negligible.

4. Compressor efficiency error analysis.

5. How to interpret uncertainty.

6. Dynamic measurement uncertainty.

8.2 MEASUREMENT UNCERTAINTY FOR MULTI-ENGINE INSTALLATIONS(SIMILAR ENGINES)

8.2.1 General

The uncertainty in performance parameters for the multi-engine aircraft for similarengines is, a function of bias limits and precision indices of the individual engines forthose performance parameters. If, for example, the parameter of interest is thrust, thetotal thrust (FT) for the aircraft is the sum of the thrust values for the individualengines. The precision index (SF T) for total thrust is the root-sum-square of the precisionindices (Si) for the individuals (assuming that the engine run-to-run variance is negligible)and where Si is the same for all engines:

2 KSF = ~S.2 KS. 2

T i= 1 11

or

SF Si v'I( (VIII-I)T

(VIII-2)Kdfi > 30

where K is the number of engines. The degrees of freedom (dO associated with aproduction engine facility would exceed 30 in almost every case. The df is calculatedfrom the degrees of freedom (dfi ) for each engine.

C!l S/) 2 (KS//df = -

~ Si4 (dll)KS i '

i=l df i

9Por a deimition of terms used in this Handbook, see the Glossary in Section IX.

129

AEDC-TR.73-5

The bias limit (B) for the installation is the sum of the bias limits (Bi ) for each of theengines:

KB = l B. = KB.

i= 1 1 1(VIII-3)

The bias limits are added rather than root-sum-squared because the bias errors for similarengines are not usually independent. (If the bias errors are independent, the bias limitsmay be root-sum-squared.) Therefore, these errors do not tend to cancel, and the limit is---best estimated as the sum of the limits.L.Ihis is consistent with the philosophy thatindependent bias limits are combined by root-sum-squa~Finally, the uncertainty forthe installation can be calculated:

(VIII-4)

where t95 is the student's "t" value for 95 percent (two-tailed) confidence and dfi timesK are the degrees of freedom.

Example of a II-nff'inno Installation

Suppose an aircraft installation consists of four, 20,000-lb-thrust engines, each withthe following reported measurement uncertainty:

Bias limit

Precision limit

±361b

±751b

df

Uncertainty

27.8

±190 lb

The precision index for the installation is calculated from Eq. (VIII-I):

SF = ±(75) y'4 = ±(75)(2)= ±150.0T

The degrees of freedom (Eq. (VUI-2)) for the precision index are 27.8 x 4 = 111.2. The biaslimit (Eq. (VIH-3)) for the installation is 4 times (±36) = ±144. The uncertaintytimlt(Eq. (VIII-4)) for the cluster is

U = ±(l4.4 + t 95 x 150) = ±[144 + 2(150)]

Since the degrees of freedom exceed 30, t95 = 2.0 is used.

±444 Ib

MEASUREMENT

The measurement uncertainty estiInate is a function of a specific measurementprocess. In this section, two different but related processes are discussed. The modelsillustrate extremes: many tests over a long period of time versus two tests over a shorttime interval. In the paragraphs that follow, the general model for many engines andmany test stands is contrasted with the model for back-to-back development testing of asingle engine on a single stand.

130

Bias PrecisionSource Limit Index Uncertainty

NBS, Interlab Standard ±O.1 ±OJ ±O.3

Interlab, Transfer Standard ±O.1 ±O.1 ±O.3Transfer, Working Standard ±O.5 ±O.5 ±1.5

Working, End Instrument ±O.5 ±O.5 ±1.5

Calibration Hierarchy Error ±O.72 ±O.72 ±2.16

Data Acquisition ±O.5 ±O.5 ±1.5

Data Reduction ±O.2 ±O.2 ±O.6

Combined Error ±O.9 ±O.9 ±2.7

AE D C-TR-73·5

Note that in the following examples the engine' hardware, instrumentation, and teststand might be identical but that the uncertainties are different because the measurementof interest is different.

8.3.1 Many Stand, Many Engine Model

The general process, which was defined in Section 1.7 and discussed throughout thisHandbook, pertains to the measurement process defined for many sets of measurementinstruments, many test stands, many calibrations, and many months of operation. Anexample is the measurementof TSFC at an engine Table XX A Measurement System with Six Error Sourcesproduction facility. Theproblem is to determine theabsolute level of performance.

The uncertainty forthis measurement process isU = ±(B + t95 S), where B isthe root-sum-square of allelemental bias limits and S isthe root-sum-square of allelemental precision indicesfor the process. For example, Table XX lists theelemental errors for a measurement system with six error sources. The root-sum-square ofthe bias limits is ±0.9. The root-sum-square of the precision indices is also ±0.9. Theuncertainty of the general process is ±2.7.

8.3.2 Single Stand~ Single Engine Model

During a gas turbine development program, many tests are devoted to evaluatingnew component designs. The objective is to obtain the most accurate determination ofthe incremental change in performance between the baseline and the new configuration.

The engine (or rig) is installed on a test stand, and a baseline calibration isperformed. Then, an engine design change is made without modification to the stand orinstrumentation. Typically, these changes may be made without removing the enginefrom the test stand. For example, the inlet guide vanes might be replaced or compressorstator vane angles adjusted. Then, the engine is tested again, and performance is measuredfor comparison with the baseline calibration. The measurement is the difference betweencorresponding performance values for the two tests. The difference should be due to onlythree causes: instrumentation precision error, engine repeatability (which has beenassumed to be negligible), and the design change. Since the repetitive tests are confinedto a single set of instruments and a single engine, the measurement uncertainty isreduced; bias errors are not considered because they will be the same for each test andwill not affect the comparison. The only errors which need to be considered are the

131

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run-to-run precision errors of the measurement system for each of the runs. For the datain Table XX, the precision index of the data acquisition is 0.5 and reduction processes is0.2 for. each run. The uncertainty must be calculated for the difference in the two runs.It is the root-sum-square of the run-to-run precision error of the data acquisition andreduction precision error for each run:

2'.12 '.10.29 = 1.523

where t95 = 2.0.

8.4 CONFIDENCE INTERVAL WHEN BIASES ARE NEGLIGIBLE OR CAN BEIGNORED

When the bias is very small (negligible) or can be ignored, Le., as in a back-to-backdevelopment test, the uncertainty parameter (D) becomes a statistical confidence interval.

The interpretation and the use of the uncertainty parameter are not changed. However,it is now defined with known exact characteristics. An uncertainty limit of six pounds forcemeans that it would be "reasonable to expect" that the true value would be within sixpounds force of the measured value. For 95 pyrcent confidence intervals, 95 percent of theintervals will contain the true value. The qualitative concept "reasonable to expect" isquantified by the confidence concept. It is not necessary to distinguish betweenuncertainty intervals with and without bias errors. The same simple interpretation ofuncertainty applies.

As an example of a confidence interval, take the following measurement data forspecific fuel consumption for a 10,000-lbf gas turbine engine:

Specific Fuel Consumption = 0.88Bias Limit = . 0.0Precision Index (S) = 0.02Degrees of Freedom = 25.2

The point estimate or unbiased estimate of specific fuel consumption is 0.88. Theuncertainty (D) is a 95 percent confidence interval estimate:

u = ± t 95 x 0.02 = ±0.041

That is the 95-percent confidence interval is 0.88 ± :0.041 or 0.839 to 0.921. In repeatedsampling, such an interval will contain the true value with 95-percertt frequency.

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8.5 COMPRESSOR EFFICIENCY ERROR ANALYSIS

8.5.1 General

Consider a compressor efficiency uncertainty analysis for the many stand, manycompressor model and the single stand, single compressor model. For each analysis, Eq.(VIII-5) is used for the calculation of adiabatic efficiency (11) of a compressor as afunction of the pressure and temperature ratios across the compressor:

where Po = inlet pressure

PI = exit pressure

To = inlet temperature

TI = exit temperature

K = ratio of specific heats

(VIII-5)

The nominal values andelemental bias limits and precision indices are listed in TableXXI. These data imply a pressure ratio of 6.5, inlet andexhaust temperatures of 530 and960C\R, and efficiency of 85percent.

8.5.2 The General Process

Many compressors aretested in many rigs with manyinstrumentation setups over aperiod of years.

The measurement uncertainty for this process is theuncertainty associated with theabsolute level of performancefor a compressor. The· intervalincludes all the bias errors andprecision measurement errorsassociated with compressor efficiency calculation. All of theerror estimates listed in TableXXI contribute to theuncertainty.

Table XXI Tabulation of the Elemental Errors

Source Bias Umit Precision Index

ToNominal, 5300

R

Thermocouples at 5300 R O.OoF to +0. 10°F 0.20oF

Reference -0. 10°F to +0.1 OOF O.lOoF

Signal Conditioning -0. 10°F to +O.lOoF 0.50oF

Data Reduction Negligible Negligible

Combining -o.14°F to +0.17°F 0.55°F

TI Nominal, 9600 R

Thermocouples O.OoF to +1.00oF 0.50oF

Reference -o.lOoF to +0.10°F O.lOoF

Signal Conditioning -0. 10°F to +0. 10°F 0.50oF


Combining -o.14°F to +l.OI°F 0.714°F

Po Nominal 14.696 psia

Transducers ±O.I% -0.015 psia 0.15% -0.022 psia

Recovery Factor Negligible Negligible

Signal Conditioning ±O.l% -Q.015 psia 0.10% -0.015 psia


Combining ±O.14% - 0.021 psia 0.18% -0.027 psia

PI Nomina195.524 psia

Transducers ±O.10% - 0.10 psia 0.15% -0.14 psia

Recovery Factor ±O.10% -0.10 psia Negligible

Signal Conditioning ±O.10% -0.10 psia 0.10% ~0.10 psia


Combining ±O.17% -0.173 psia 0.18% -0.17 psia

Degree of freedom is assumed> 30.K is ratio of specific heats, Nominal, 1.39, errors negligible.

133

Source Bias Limits Precision Index

To BTo =-0.14°p Br =0 17°p STo =O.5Sopo .Tl BTl =-0.14°p Bt =1.01°p STl = 0.714°p

0

Po Bpo =±O.14% or 0.021 psia Spo =0.18% or 0.027 psiaPl Bpl =±O.17% or 0.173 psia SPl =0.18% or 0.17 psia

AEDC-TR-73-5

For each of the four values To, T1, Po, and Pl , the elemental bias limits areroot-sum-squared and the elemental precision indices are root-sum-squared. Table XXIIpresents a summaryof the combined bias Table XXII Summary of Errorslimits and precisionindices from TableXXI.

The errors inTo, TI, Po, and Plare propagated toefficiency using theTaylor's series meth-od described in Appendix B. The appropriate calculations for the propagation are

S 2TJ (~s )2aT T

o 0 (VIII-6)

B 2 =TJ

(VIII-7)

where

K-l

-To [(P/P o)K -~(T 1 - To) 2

-1

-(~)&/Po)K(P/Po~(T l/T 0) - I

the rate of change of'Y1 with respect toTo

the rate of change of 'Y1 with respect toTl

the rate of change of'Y1 with respect toPo

the rate of change of'Y1 with respect toPl

By substituting into Eqs. (VIII-6) and (VIII-7), the' precision index and the upperand lower bias limits are calculated.

134

II

AE D C-TR-73-5

S 2 [(0.0036)(0.55)]2 + [(-0.00198)(0.714)]2 + [(--0.0398)(0.027)]2 + [(0.0061)(0.17)]2TJ

0.00000815

STJ 0.002854

(BTJ_)2 = [(0.0036)(--0.14)] 2 + [(-0.0020)(-0.14)] 2 + [( -0.0398)(0.021)] 2 + [(0.0061)(0.173)] 2

0.00000214

B - 0.00146TJ

(B +)2 [(0.0036)(0.17)] 2 + [( _ 0.0020)(1.01)] 2 + [(0.0398)(0.021)] 2 + [(0.0061)(0.173)] 2TJ

0.00000626

B + 0.0025TJ

Bi1 is assigned a negative value since the lower limits in Table XXI are all negative.

The propagated values to efficiency are S = 0.0029 for the precision index, and B- =-0.0015 and B+ = 0.0025 for the bias limits. The error in specific heat parameters (K) isconsidered negligible.

The uncertainty for the general model is the interval contained between U- and U+where:

(VIII-B)

(VIII-9)

In this case, t95 = 2.0 since all the degrees of freedom are greater than 30.

u- = -0.0015 - 2(0.0029) -0.0073

u+ = +0.0025 + 2(0.0029) 0.0083

For the nominal values of Table XXI, the uncertainty interval would be 85% - 0.0073 to85% + 0.0083 or 84.27% to 85.83%.

8.5.3 Single Stand, Single Compressor Process

For a back-to-back development process, two tests are performed on a single standwith a single compressor. No changes are allowed to the stand system or data recordingequipment. The process would involve testing the compressor to establish a baselineefficiency value, changing the compressor configuration and retesting to determine a newefficiency value and to determine the delta from the baseline efficiency value.

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AEDC·TR·73.5

The uncertainty interval for these tests is a function of the precision error only ofthe measurement system. The original table of elemental errors (Table XXI) is nowreduced to Table XXIII.

Table XXIII Elemental Errors

Parameter To, OF TI' OF Po, psia ' PI, psia

Signal Conditioning 0.5 0.5 0.015' 0.1

Precision Index Reference 0.1 0.1 --- ---Root-Sum-Square 0.51 0.51 0.015 0.1

The root-sum-squareerrors are propagated toefficiency' using Taylor'sseries methods describedabove and in Appendix B.The measurement of concern is the delta betweentwo tests. Therefore, theuncertainty must be propagated to the delta value:

where 'Yl2 is the efficiency for the second run and 'Yll is the efficiency for the fust run. Thepropagation for the precision term SA'Yl is:

(VIII-I 0)

~easurement Process Uncertainty Interval

General ~easurement Process + 3.7(for Intercompany Comparisons)

Back-to-Back Testing + 2.0~easurement Process

The uncertainty resulting from the values of Table XXIII is 0.0065.

8.6 HOW TO INTERPRET UNCERTAINTY

Uncertainty is a function of the measurement process as discussed in Section 8.3. Adifferent definition of the process would significantly change the uncertainty. Table XXIVlists the uncertainty values for the many stand, many engine model and for the single stand,single engine model. These values are significantly different, yet both are based on theelemental errors listed in Table XXI.

Table XXIV Uncertainty Values for Two Processes Uncertainty, then, is a functionof the measurement process. Itprovides an estimate of the largesterror that may reasonably be expected for that measurement process(Fig. VIII-I).

Errors larger than the uncertainty should rarely occur. On re

peated runs within a given measurement process, the parameter values should be withinthe uncertainty interval. These differences might look like Fig. VIII-2. Run-to-rundifferences between corresponding values of Parameter A should be less than theuncertainty for A.

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AEDC-TR-73-5

Measurement

.....--- Largest Negative Error-(B + t 9SS)


+(B + t 9SS)

Measurement Scale

Range of ±t9SS104---- - B ------....-----Precision ----11IO....---+ B -------IIiII't

Error

.....--------------- Uncer t a i nty 'I nterva I ---------------.....-t(The true value should fall within this interval)

Fig. VIII-' Measurement Uncertainty

-U ........----------------

o

o

oo

Run Number

Parameter A

oo

u

Fig. VIII-2 Run-to-Run Differences

Average ofPast Tests

If the difference to bedetected in an experiment is ofthe same size or smaller thanthe proj ected uncertainty,corrective action should betaken to reduce the uncertainty. Therefore, measurement. uncertainty analysisshould always be done beforethe test or experiment. Thecorrective action to reduce theuncertainty may involve (l)improvements or additions to the instrumentation, (2) selection of a different function toobtain the parameter of interest, and/or (3) repeated testing. Cost and time will dictatethe choice. If corrective action cannot be taken, the test should be cancelled as there is ahigh risk that the real differences will be lost in the uncertainty interval (undetected). Ifthe measurement uncertainty analysis is made after the test, the opportunity forcorrective action is lost, and the test may be wasted.

8.7 DYNAMIC MEASUREMENT UNCERTAINTY

The same basic measurement uncertainty model may be applied to time-varyingdata. However, there are added complex problems to solve, and the services of astatistician are recommended. Some of these problems are

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AEDC·TR-73-5

1. Time LagsDifferent instrumentation will have different time lags which cause seriousproblems in determining time-variant parameters. For example, thrustspecific fuel consumption (TSFC) is based on fuel flow and on thrustdeterminations. During a transient, if the measured value for one lagsbehind the other, the ratio, TSFC, will be in error.

2. Uncertainty Varies with Parameter LevelThe uncertainty of the measurements will probably change as the level ofthe parameter changes. This will be hard to predict because someinstruments have errors which are constant, independent of level (error is aconstant percentage of full scale, for example); other instrument precisionerrors and biases vary as the level of the measured parameter varies (erroras a percentage of point). Therefore, the uncertainty will be a combinationof these two types of error and will neither remain a constant percentagenor change as a constant percentage of point.

3. Shifting Flow ProfilesFlow profiles are usually considered fixed at steady-state points. Duringtransients, the profile will often shift or change. The uncertainty isincreased by this added variation.

4. Autocorrelation between MeasurementsTime-variant measurements on gas turbines will usually be highly related intime (autocorrelated). The degree of autocorrelation will have a significanteffect on the uncertainty of the performance parameter.

5. Number of Probes, Location of Probes, Sampling RateIf the parameter to be measured involves extreme values like inletdistortion and burner temperature pattern factor, the uncertainty will behighly dependent on the number and the location of the probes. If theparameter involves frequency, the time rate of sampling will be significantand the uncertainty will vary as a function of both sampling rate andfrequency.

6. OutliersOutliers in time-variant data are much more difficult to detect and flagbecause of the variation in the level of the parameter. This could result inmore outliers being included as good data because they appear to bevariations in the parameter. Therefore, for time-variant data, an outlierdetection technique should be used very carefully.

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SECTION IXGLOSSARY

1. Accuracy - The closeness or agreement between a measured value and a standard ortrue value; uncertainty as used herein, is the maximum inaccuracy or error thatmay reasonably be expected (see measurement error).

2. Average Value - The arithmetic mean of N readings. The average value is calculatedas:

x average value

NIx.i=l 1

N

3. Bias (B) - The difference between the average of all possible measured values andthe true value. The systematic error or fixed error which characterizes everymember of a set of measurements (Fig. IX-I).

True Value

Average

1oE----Bias------Il-..-~

Fig. IX-' Bias in a Random Process

4. Calibration - The process of comparing and correcting the response of aninstrument to agree with a standard instrument over the measurement range.

5. Calibration Hierarchy - The chain of calibrations which link or trace a measuringinstrument to the National Bureau of Standards.

6. Correlation Coefficient - A measure of the linear interdependence between twovariables. It varies between -1 and +1 with the intermediate value of zeroindicating the absence of correlation. The limiting values indicate perfectnegative (inverse) or positive correlation (Fig. IX-2).

139

AEDC·TR·73.5

.... . .... '.I, ',' " .' •....

, .r ... 0.0

r ... 1.0

r· ... 0.6

r ... 0.0

...........I, • II

. ..,.'

r ... 0.8

r ... -1. 0

fig. IX·2 Correlation Coefficients

7. Coverage - A property of confidence intervals with the connotation of including orcontaining within the interval with a specified relative frequency. Ninety-fivepercent confidence intervals provide 95-percent coverage of the true value. Thatis, in repeated sampling when a 95-percent confidence interval is constructedfor each sample, over the long run the intervals will contain the true value 95percent of the time.

8. Degrees of Freedom (dO - A sample of N values is said to have N degrees offreedom, and a statistic calculated from it is also said to have N degrees offreedom. But if k functions of the sample values are held constant, the number

N -of degrees of freedom is· reduced by k. For example, the statistic.~ (Xi - X)2,

_ pIwhere X is the sample mean, is said to have N - 1 degrees of freedom. Thejustification for this is that (a) the sample mean is regarded as fixed or (b) innormal variation the N quantities (Xi _X) are distributed independently of Xand hence may be regarded as N - i independent variates or N variatesconnected by the linear relation ~(Xi - X) = O.

9. Elemental error - The bias and/or precision error associated with a singlecomponent or process in a chain of components or processes.

10. Estimate - A value calculated from a sample of data as a substitute for an unknownpopulation constant. For example, the sample standard deviation (S) is theestimate which describes the population standard deviation (0-).

11. Joint Distribution Function - A function describing the simultaneous distribution oftwo variables. The cumulative probability distribution for 2 variables.

12. Laboratory Standard - An instrument which is calibrated periodically at the NBS.The laboratory staiidardmay also be called an interlab standard.

140

AEDC-TR-73·5

13. Mathematical Model -A mathematical description of a system. It may be aformula, a computer program, or a statistical model.

14. Measurement Error - The collective term meaning the difference between the truevalue and the measured value. Includes both bias and precision error; seeaccuracy and uncertainty. Accuracy implies small measurement error and smalluncertainty.

15. Multiple Measurement - More than a single concurrent measurement of the sameparameter.

16. NBS - National Bureau of Standards. The reference or source of the true value forall measurements in the United States of America.

17. Parameter - An unknown quantity which may vary over a certain set of values. Instatistics, it occurs in expressions defining frequency distributions (populationparameters). Examples: the mean of a normal distribution, the expected valueof a Poisson variable.

18. Precision Error - The random error observed in a set of repeated measurements.This error is the result of a large number of small effects, each of which isnegligible alone.

19. Precision Index - The precision index is defined herein as the computed standarddeviation of the measurements.

s

_ 2~ (x. - X)i= 1 1

N-lusually, but sometimes S = ~~ s2

20. Proving Ring - Laboratory standard for force measurements.

21. Sample Size (N) - The number of sampling units which are to be included in thesample.

22. Standard Deviation (a) - The most widely used measure of dispersion of afrequency distribution. It is the precision index and is the square root of thevariance: S is an estimate of a calculated from a sample of data.

23. Standard Error of Estimate - The measure of dispersion of the dependent variable(output) about the least-squares line in curve fitting or regression analysis. It isthe precision index of the output for any fixed level of the independentvariable input. The formula for calculating this is

i~l (v OBS - VCAL )2N-K

for a curve fit for N data points in which K constants are estimated for thecurve.

141

Its form is

AEDC-TR-73-5

24. Standard Error of the Mean - An estimate of the scatter in a set of sample meansbased on a given sample of size N. The sample standard deviation (S) isestimated as

Then the standard error of the mean is S/.JF[ In the limit, as N becomes large,the estimated standard error of the mean converges to zero, while the standarddeviation converges to a fixed non-zero value.

25. Statistic - A parameter value based on data. X and S are statistics. The bias limit, ajudgment, is not a statistic.

26. Statistical Confidence Interval - An interval estimate of a population parameterbased on data. The confidence level establishes the coverage of the interval.That is, a 95-percent confidence interval would cover or include the true valueof the parameter 95 percent of the time in repeated sampling. .

27. Statistical Quality Control Charts - A plot of the results of repeated sampling versustime. The central tendency and .upper and lower limits are marked. Pointsoutside the limits and trends and sequences in the points indicate non-randomconditions.

28. Student's "t" Distribution (t) - The ratio of the difference between the populationmean and the sample mean to a sample standard deviation (multiplied by aconstant) in samples from a normal population. It is used to set confidencelimits for the population mean.

29. Taylor's Series - A power series to calculate the value of a function at a point inthe neighborhood of some reference point. The series expresses the differenceor differential between the new point and the reference point in terms of thesuccessive derivatives of the function.

f(X) _ f(a) = 1'=i,-1 (X_a)T f(r)(a) + R1'=1 r! . n

where f(r)(a) denotes the value of the rth derivative of f(x) at the referencepoint X = a. Commonly, if the series converges, the remainder Rn is madeinfinitesimal by selecting an arbitary number of terms.

30. Traceability - The ability to trace the calibration of a measuring device through achain of calibrations to the National Bureau of Standards.

31. Transducer - A device for converting mechanical stimulation into an electricalsignal. It. is used to measure quantities like pressure, temperature, and force.

32. Transfer Standard - A laboratory instrument which is used to calibrate workingstandards and which is periodically calibrated against the laboratory standard.

142

33. True Value - The reference value defined by the National Bureau of Standardswhich is assumed to be the true value of any measured quantity.

34. Uncertainty (U) - The maximum error reasonably expected for the definedmeasurement process: U = ±(B + t95 S).

35. Variance (a2 ) - A measure of scatter or spread of a distribution. It is estimated by

S2 = ~ (Xi - X)2 from a sample of data. The variance is the square of theN-l

standard deviation.

36. Working Standard - An instrument which is calibrated in a laboratory against aninterlab or transfer standard and is used as a standard in calibrating measuringinstruments.

143

APPENDIXESA. PRECISION INDEX FOR UNIFORM DISTRIBUTION

OF ERRORB. PROPAGATION OF ERRORS BY TAYLOR'S SERIESC. ESTIMATES OF THE PRECISION INDEX FROM

MULTIPLE MEASUREMENTSD. OUTLIER DETECTIONE. TABLES

145

A E 0 C-T R -73-5

AEDC-TR-73-5

APPENDIX APRECISION INDEX FOR UNIFORM DISTRIBUTION OF ERROR

The precision index for a uniform distribution of error is easily calculated byconsidering the definition of the variance:

2fb 2(1 = (x -,.,.) p (x) dxa

(the general formula for the variance).

For a uniform distribution (Fig. A-I) between the limits of a and b, the formula is

(12 =lb ( _a + b)2 1 dx 2 (b _ a) x

a

where (a + b)/2 is the mean (Jl) of the uniform distribution.

(12 =jb (x2 _ (a 82

+ 2:b + b2

) 1+ b) x + (b _ a) dx

1(b - a)

(a + b) [b2- a 2] + a

2+ 2ab+ b2 rb aJ

(b - a) 2 4(b - a) l

2(b - a)

12_ r (b - a)2orO'-\j 12

When a = -1 and b = +1, 0' = IIi ·0 - (- 1 •0 )J~ = {I z O. 5 77\j 12 \J 3

when a = -1/2 and b = +1/2, (1

------....--.+--...,Fig. A-1 Uniform Distribution

of Error

147

p(x)

_1_b - a

I a+b11""-I . 2

I

IIIIIIIIII

AE DC-TR -73-5

APPENDIX BPROPAGATION OF ERRORS BY TAYLOR'S SERIES

GENERAL

The proofs in this section are shown for two- and three-variable functions. Theseproofs can be easily extended to functions with more variables, although, because of itslength, the general case is not shown here.

TWO INDEPENDENT VARIABLES

If it is assumed that response Z is defined as a function of measured variables (x andy), the two restrictions that must be considered are

1. Z is continuous in the neighborhood of the point (/lx, /ly). Both x and ywill have error distributions about this point and the notation (/lx and J.ly)indicates the mean values of these distributions.

2. Z has continuous partial derivatives in a neighborhood of the point (/lx,My).

These conditions are satisfied if the functions to be considered are restricted tosmooth curves in a neighborhood of the point with no discontinuities Gumps or breaks inthe curve). The Taylor's series expansion for Z is

(B-1)

az az .where ax and ay are evaluated at the pOInt (Mx, My).

(B-2)

a2 z a2 zwhere -- and -- are evaluated at (8 1,82) with 81 between x and Mx, and 82 betweenax 2 ay2

y and My.

The quantity R2, the remainder after two terms, is not significant if either:

1. (x - Mx) and (y - My) are small

a2 z a2 z2. The second partials -- and --2 are small or zero. These partials are zeroax2 ayfor linear functions.

149

AEDC-TR-73·5

By assuming R2 to be small or zero, Eq. (B-1) becomes

or

Z ~ II + dZ (x ) + dZ ( ),....Z oX - J..Lx Oy y - J..L y

• OZ ( ) dZ (Z - ~Z = dX x - ~x + OY Y - My)

(B-3)

(B-4)

By defining JJ.z as the average value of the distribution of Z, the difference (Z - JJ.z )is the difference of Z about its average value. This difference may be approximated by(Eq. (B-4))

(B-5)

where the partials are evaluated at the point (JJ.x, JJ.y).

The variation in Z is defined by

UZ2 =_ f:(Z _ II )2 p dZJ' ,....z Z

where pz is the probability density function of Z. Therefore,

2 _(1[OZ dZ ] 2 "(1 ZJ.. ax (x - IJ.x ) + dY (y - }J.y) Pxy dxdy

"ff[~~(x -~x)]2PXY dydx 1f[~~(Y -~y)]2PXY dxdy

+ g[~~(x - ~x>][~~(y - ~y)] Pxy dxdy

(B-6)

(B-7)

where' Pxy is the joint distribution function of x and y. Integrating the first term of Eq.(B-7) with respect to y and the second term of Eq (B-7) with respect to x gives

jjrj az dZ '.+ 2 -s- (x - }J. ) ~(y - M')p dxdyOX x oY Y xy

(B-8)

If JJ.x and JJ.y are the means of the distributions of x and y, then define the following:

a~ =fix - JJx)2~xdX (B-9)

150

AE DC-TR-73·S

- 1J)p dxdyy xy

(B-lO)

(B-1 1)

where Pxy is the coefficient of correlation between x and y. Combhring the definitionsand Eq. (B-8) gives

(B-12)

If x and y are independent variables, then P =0 and

(B-13)

THREE INDEPENDENT VARIABLES

If it is assumed that Z is a function of variables x, y, and w, two restrictions mustbe considered:

1. Z is continuous in a neighborhood of the point (Pox, Poy, Pow)

2. Z has continuous partial derivatives in a neighborhood of (Pox, Poy, Pow)

If these restrictions are satisfied, then the Taylor's series expansion for Z in the vicinityof (Pox, Poy , Pow) is

where

(B-14)

(B-15)

These second partials are evaluated at a point (h, (}2, (}3 defined so that (}1 is betweenPox and x, (}2 is between Poy and y, and (}3 is between Pow and w. The same restrictionsapply to R2 as defined for two-variable functions.

By assuming R2 to be small or zero, Eq. (B-14) becomes

Z - 1J ~ ~z (x - J.4. ) + ~z (y - J.4. ) + ~z (w - J.4. )Z ox x oY Y oW W

where the partials are evaluated at the point (Jlx, Poy, Pow)·

151

(B-16)

A E D C-T R -73-5

The variation in Z is defined by

.a~ =JJJ[~~(X

=fll[~~(X

,..2Z

:;;fiz H)2 dZv - ,...Z P Z

where pz is the probab~lity density function of Z. Therefore,

az aZ ']2- J.lx ) + OY (y - J.l y ) + OW (w - J.l w) Jx,y,w

- J.l x )J2 Px 'y 'w dwdydx + •.•

+ 2jjrrraZ azJJ ax ay (x - J.L x ) (y - J.Ly) Px' Y 'w dwdxd y + ...

(B-17)

dxd ydw (B-18)

(B-19)

where Px,y,w is the joint distribution function of x, y, and w. Integrating 'in the properorder produces these results:

Therefore,

(B-2 1)

If x, y, and ware independent variables, then Pxy =Pxw =Pyw =0 and

(B-22)

MONTE CARLO SIMULATION

To determine the restrictions that must be placed on applications of the method ofpartial derivatives, a Monte Carlo Simulator was designed to provide simulation checks forthe computation of various functions. Comparative results are listed in Tables B-1. andB-2.

Table B-1 contrasts the results of the Monte Carlo simulation of the functionstabulated, column (7), with the estimates using Partial Derivatives, column (6). Onethousand functional values were obtained in each simulation. Column (l) identifies thefunction simulated and column (2) gives the number of the simulation run. Column (3),Theoretical Input, includes the parameters of the populations from which the random

152

AEDC-TR·73·5

Table B-1 Results of Monte Carlo Simulation for Theoretical Input (ax 2 , IJ.x, ay 2 , IJ.y)

(1) (2) (3) (4 ) (5 ) (6) (7)

Function Simulation Theoretical Method of Method of Input Variance Obse rv£:dRun Number Input Partials Partials Corrected for Variance

2~x

2 Estimated Estimated Nonindependence (SimulatorO'x O'y ~y Variance Variance (Method of Results)

(Theoretical) (Ac tua1 Inpu t) Partials)

x + y 1 1.0 10 4.0 20 5.0 4 .. 9477 4.8496 4.85672 1.0 10 4.0 20 5.0 4.9186 4.8435 4.85063 1.0 10 4.0 20 5.0 5.0786 4.9493 4.95644 1.0 10 4.0 20 5.0 5.1639 5.2444 5.2515

x - y 1 1.0 10 4.0 20 5.b 4.9477 5.0358 5.04102 1.0 10 4.0 20 5.0 4.9186 4.9937 4.98853 1.0 10 4.0 20 5.0 5.0786 5.2079 5.20284 1.0 10 4.0 20 5.0 5.1639 5.0834 5.0782

(x) (y) 1 1.0 10 4.0 ~o 800.0 792.81 773.27 768.632 1.0 10 4.0 20 800.0 794.33 779.29 797.483 1.0 10 4.0 20 800.0 802.28 776.41 775.784 1.0 10 4.0 20 800.0 867.67 883.85 883.38

x/y 1 1.0 10. 4.0 20 0.005 0.0050 0.0051 -0.00542 1.0 10 4 •.0 20 0.005 0.0050 0.0051 0.00543 1.0 10 4.0 20 0.005 0.0050 0.0052 0.00554 1.0 10 4.0 20 0.005 0.0054 0.0053 0.0057

numbers were drawn. Column (4) lists the method of partials estimates of variance forthe function based on the theoretical input (column 3). Column (5) lists the estimates ofvariance for the function calculated using the method of partial derivatives from theobserved variation of the variables x and y. Column (6) gives column (5) corrected forthe observed correlation between the pairs of x, y input values. The correction factor is:

where p is the observed correlation between paired values of x and y, ax 2 and ay 2 are

the observed variances of x and y, and ~~ and ~~ are the partial derivatives of the

function Z. Column (7) lists the simulator results for the function (column 1) for 1000data points.

Table B-2 Results of Monte Carlo Simulation for Theoretical Input IJ.xjt a~i

(1) (2) (3) (4) (5)Function Number of 'lheoretical Estimated Parameters Simulation Results

Z Simulations Input (Method of Partials)2

IIoZ2 IIoZ

2IIox . tTx . tTZ tTZ

~ ~

(xlx2) /x3 2 20 1.0 20 3.00 20.2 2.5620.6 3.24

(xl x2) / (x3x4x5) 20 1.0 0.05 -5 0.0505 -53.12 x 10 3.6 x 10

(x1x2x3X4)/(x5x6x7) 2 20 1.0 20 7.00 20.04 8.4120.25 8.41

(x.X2XJ)~i~4x~ 20 1.0 1.25 x 10-4 3.52 x 10.10 -4 4.0 x 10-101.29 x 10

(i~lxi)I'(X7X8X9) 2 20 1.0 8000 1.44 x 106

8150 1.69 X 106

8300 1.82 x 106

153

A EDC-TR.73.5

Columns (1) through (3) of Table B-2 present the input to the Monte CarloSimulator. The theoretical imput column (3) shows the parameters of the population ofrandom numbers that were used to produce the functional values. Column (5)summarizes the results of the simulation. These results may be compared with theestimates from the method of partials, column (4).

Simulation results have shown that the method of partial derivatives is most accuratefor functions involving sums and differences of the observed variables. For thesefunctions, if the variables are mutually independent, the Taylor's series is exact for anymagnitude of error in the measured parameters. If the variables are not mutuallyindependent, a correction factor can be COI!!puted that will ensure exactitude of the

method. (The correction factor (2px y ax ay az ,aZ) is the third term in Eq. (B-12). If Px y~x,ay

is not zero, this term should be included in estimating az 2. From data, Pxy may beestimated with

Sr = -2£i..

S Sx y

2: (x. -~) (y. -y). ~ ~

~ - 2 - 2(x. -x) ~ (y. -y)1. 1.

where n pairs of observations are available and x and yare the average of the Xi and Yi

values, respectively.)

Close approximations can be made for errors that exist in functions involvingproducts and quotients of independently varying observed values if the ratio of measurederrors to their respective nominal values is small (less than 0.1). The approximationimproves as measured errors decrease in relation to their nominals. For all of thefunctions examined involving two or more independent variables, the approximation iswithin 10 percent of the true error. The simulation results are summarized in Tables B-1and B-2.

Table B-3 ·shows the Taylor"s formula for several functions. In addition, the Taylor'sformula for the coefficient of variation is also listed. The coefficient of variation is easilyconverted to a percentage variation by multiplying by 100.

154

AEDC-TR-73-5

Table B-3 Error Propagation Formulas

Function Taylor's Formula Coefficient of VariationFormula

\v = f (x,y)

w = Ax + By

1w

y

xw =--

x + Y

xw=--1 + x

8 2 :::: 82

-:Lw4

y

(S 2 (X S Y82 - Y x

- (x+y) 2) +~w

82 ::::8 2

xw

(l + x) 4

V 2V2 :::: _x_

w 4

( V Jv2 _ 2Lw - .lux

8x= -=x

vx

8 2~ _x_

4x

w = xy

2w x

1/2w x

w ln x

kxaybw =

where:

v = ~y y

SV = :'w w w

155

AEDC·TR.73-5

APPENDIX CESTIMATES OF THE PRECISION INDEX FROM MULTIPLE MEASUREMENTS

INTRODUCTION

The derivatives, proofs, and examples of this section are presented for measurementschemes using two instruments measuring the same parameter. These instruments are readsimultaneously and the difference between the readings is analyzed statistically toestimate instrumentation precision index. The proofs and derivations can easily beextended to instrumentation setups using more than two instruments by considering allindependent combinations of instrument pairs. Instrumentation pairs which have a fixedconstant locational bias between them can also be analyzed to provide estimates ofprecision index.

DISCUSSION

In any measurement system, if the parameter which is being measured is a constant,the precision error of the measurement instruments is easy to i~entify. The variance (S2)of the measurements (Xi) about the average measurement (X) provides an unbiasedestimate of the variance of the instrument:

(for N measurements)(C-l)

In the usual case, the measured parameter is not constant. Then the variation of thereadings is increased by the variation of the parameter, and any directly computedstatistic is subjected to large errors. However, multiple instruments can be analyzed toestimate the precision error in the measurement system.

For example, if two instruments are measuring a parameter X, then an appropriatemathematical model for each reading would be Xi + €ij where Xi is a typical parametervalue and €ij is the corresponding precision error for the jth instrument. (For simplicity,bias error is ignored as it has no effect on these methods.) A series of such readings isillustrated in Table C-l.

Table C·, Multiple Measurements of a Parameter

TrueValue Instrument One Instrument Two

Xl X-l + tIl Xl + El2

X2 X2+ E

2l X2 + E22

X3 X3 + E31

X3 + t 32

• • •• • •• • •X X + t n1

X -r E1l2n n n

157

A ED C-T R-73-5

This list is provided by making simultaneous readings of instruments one and two.By subtracting the reading of instrument one from instrument two, a column ofdifferences can be produced which is independent of the variation of the parameter(Table C-2).

Table 'C·2 Multiple Measurement Difference

Instrument One Instrument Two Average Difference

Xl + Ell Xl + E12 Xl +Ell + E12

Ell - E12 = &112

E2l + E22X2 + E

21X

2+ E

22 X2 + 2E2l

- E22 = &12

E3l + E32X3

+ E31 X

3+ E X

3 + E31- E

32= &1

332 2

E - E = Anl n2 n

The variation of these differences (SA) provides an unbiased estimate of the precisionerror of the average reading of the two instruments. That is, SA can be used to estimateSreading based on Eq. (C-l).

s~ n - 1

(C-2)

where A is the average difference between the meters and n is the number of differences.It can be proved that the estimate of the precision index of the average reading is then

(C-3)

4 (n - 1)

158

AEDC.TR.73·5

This is based on the assumption that the two meters have independent precision errors.The formulation is derived from Taylor's Series expansion in Appendix B, for precisionerrors in calculated values. For example, if a single pair of multiple readings is made, theerror is (€i1 + €i2 )/2 if the average of the two readings is recorded. The precision indexof this average value may be estimated using the Taylor's series expansion:

Thus

(C-5)

and

S (til : fi~) ~

In the same manner, the precision index of €n - €i2 (Eq. (C-2)) can be shown to be:

(C-6)

a(til - f i 2) S

c)E i1 fil

or

(C-7)

(C-8)

By combining Eqs. (C-2), (C-6), and (C-8), the precision error estimate for the averagereading of Table C-2 is

~ (~. _ ~ )21 1

4(n-l)

159

1 1S = -2 s~

2 ~il - f i2)

(C-9)

AEDC-TR-73-5

The method described in the preceding discussion does not include long-term drifts,calibration errors, and other precision errors that do not vary from data point-to-datapoint, from stand-to-stand, and from calibration-to-calibration. A new mathematicalmodel must be developed based on an engineering analysis including these effects.

To illustrate the method, the mathematical model derived for a tutbojet engine willbe used. Again, X is the parameter being measured; then

Re ad i rig" = X .• n + a. + 13.. + E•. IJL ~J.t ~ ~J ~JL

is the mathematical model of reading Q of a typical instrument during run j and sincecalibration i. The model assumes a within test (short-term) component of precision error(€ij,&), a test-to-test (long-term) component or precision error (f3ij) which is constantwithin a test and a calibration-to-calibration error (ai) which is constant within eachcalibration and within test within each calibration.

Table C-3 illustrates a series of readings of a multiple instrumented parameter,assuming this model and the differences of the simultaneous readings of instruments oneand two. Notice that, within each run, the run-to-run and calibration-to-calibrationcomponents are constant. Between the averages of each run, the within run componentsare reduced and the calibration-to-calibration components are constant allowing estimatesof run-to-run components. Finally, the between two calibrations, the within run, andrun-to-run components are reduced allowing an estimate of calibration-to-calibrationerror. The method of analysis is that of a one-way nested analysis of variance.

The analysis can be completed· using the formulas of Table C-4. The last column ofthis table provides the estimates of the precision index for calibration-to-calibration error,run-to-run error, and within run error. The third column, mean square, is calculated bydividing the value of the sum of squares (column I) by the proper degrees of freedom(column 2).

The estimate of the total preCISIon index, for a multiple instrumented parameter(average of two measurements) on a particular run and calibration, would be theroot-sum-square of the three estimates: within test precision index (Sw t), test-to-testprecision index (Stt), and calibration-to-calibration precision index (Sec); that is,

2 S 2 S2 S 2ST = wt + tt + cc

The degrees of freedom for this estimate can be calculated from the degrees offreedom for each estimate using the Welch-Satterthwaite formula.

160

Table C-3 Differences in Readings of Multiple Instruments

lSI D!STJUDPl

Xlll + all + ~lll + f llll

c x1l2"+ all + ~lU + f 1121~

2BD D!StRUHENt

Xlll + a 12 + ~112 + t ll12

X l12 + a 12 + ~112 + El122 •

punmgs

all + ~lll + f lill - (a12 * ~112 + Ell12)

aU + ~Ul + f U21 - (aU + ~U2 + E U22]

~

- 4 1U

-4U2

.u •••••••••••• •..... XUn + all + ~lll + f Unl

...III

U

~1!N

X12l + all + ~121 + t 21U

X122 + all + ~Ul + E2121

XUn + aU + ~121 + E21nl

x Un + a 12 + BU2 + EUn2

X12l + aU + B122 + f 2U2

X122 +a12 + ~122 + E2122

X12n + a 12 + ~122 + E21n2

au + ~lU + f llnl - [aU + 8il2 + funi)

all + ~121 + E2l11 - (a12 +~122 + f 2112]

aU + ~Ul + f 2121 - ~U + ~U2 + f 2122)

au + ~121 + f 21nl - [aU + ~122 + f 21n2]

-4 Un

- 4 2U

-4212

- 4 2ln

0\-

XIKI + aU + 'lKl + f KUl

§ X1K2 + all + ~lKl + f K12lIll::.c ••••••••••• ••~

X1Kn + all + ~lKl + EKlnl

X2U + a 21 + ~211 + EUU

§ X212 + a 21 + ~211 + f U21Ill::.u •••••••••••• •

.!lX2ln + a 2l + ~211 + f l2nl

XIKI + a 12 + ~1K2 + f KU2

XlK2 + a 12 + ~1K2 + f K122

XUCn + a 12 + ~1K2 + EKln2

X2U

+ a22

+ B212 + f l212

X2l2 +a22 + ~2l2 + EU22

X21n + a 2i + ~2l2 + E12n2.

au + ~lKl + f KUl - [a12 + ~1K2 + fUU]

aU + ~lKl + f K121 - [al2 + ~lK2 + f K122]

...... ' .au + ~lKl + f Klnl - [a12 + ~lK2 + EKln2]

CIl21 + 8.211 + t l'211 - [a22 + 8212 + fUU]

CIl 21 + 8.211 + f I22l - (a 22 + ~2l2 + E1222]

CIl 2l + 8.2u + f 12nl - [a22 + ~2l2 + E12n2)

-4Kll

- 4 K12

- 4 Kln

- 4 121

- 4 122

- 4 l2n

X2Kn + a 21 + ~2Kl + t K2nl

X2Kl + a 2l + ~2Kl + f K2U

§ X2K2 + a 2l + 82Kl + f K22lIll::.t:: ••••••••••••.u:.:

N....III

U

§Ill::

1!N

X22l + a 2l + ~22l + f 22U

X222 + a 2l + ~22l + E222l

X22n

+ a2l + 8221

+ E22nl

X22l + a22

+ tJ222

+ f 22l2

X222 +a22

+ 8.222 + f 2222

X22n + a 22 + ~222 + E22u2

X2Kl + a 22 + ~2K2 + t K2l2

X2K2 + a 22 + ~2K2 + f K222

X2Kn + a?2 + ~2K2 + f K2n2

a21 + ~22l + f 22U - [a22 + ~222 + f 2212)

a21 + 8.221 + f ~22l - [a22 + ~222 + f 2222]

CIl2l + 8.221 + f 22nl - [a22 + 11222 + f 22n2)

an + 8.2Kl + f K2U - [a 22 + 11 2K2 + f K2l2 )

a 21 + 8.2Kl + EK221 - [a 22 + 112K2 + f K222)

a 21 + ~2Kl + f K2nl - [a22 + 82K2 + f K2n2 )

- 4 221

- 4 222

-422n

- 4 K2l

"'4K22

-A K2n

»mo()

~:D~(,.)

crt

0\N

Table C-4 Analysis of Precision Error from Multiple Instrument Differences

Source Sum of Degrees Mean Varianceof Squares of Square Estimates

Variation Freedom

t. l.

BetweenM

2 T2 2 - 2 - 2 ,,2 =,,2 =

M2 - "0 - N2"1E (Ti INi ) - N" M-1 M2 -"0 + N2"1 + N3"2 N

3Ca1s Cal 2

i=l Cal

K. M2

M l. 2 2 2 - 2 ,,2 =,,2 =M1 -"0Between E E (Ti/Nij ) - E (T

iIN

i) R-M M1 ""0 + N1"1Runs

i=l j=l i=1Run 1 N1Run

M K. ;: 2 11 K.l. l.

2 2 2 2Within E E .1 - E E (Ti/Nij ) N-R MO .. "0 "Within ="0Runsi=l j"'l 1=1

ijli=l j-1

MK. f; 2

T2l.

Total E E .1 N-li=l j=l 1=:1

ij1 -N"

M MK

i MK

i ;:T '" E T '" E E T '" E E .1 ij1i-I i i"'l j"'l ij i-I j-1 1-1

M M Ki K

N '" E N = E E N. jR- E K

ii"'l i i"'l j"'l l. i-I

[. - M C' .~/.,)] I(R-M)N1 '" ~ ~ ~1

[M (t ~/.,)- M x. ]N = E E t N~.IN I(K-l)

2 . i"'l j=l i"'l j=l J

.3" ~2_M .';)I(<M-U.)E

i=l

»mo(")

~:0

~Wen

A ED C-T R-73-5

APPENDIX DOUTLIER DETECTION

GENERAL

x~ Spurious Dato

---X------,------x- .-l( x l( l( x X lC X Expected

l( l( X lC lC lC lC lC lC lC Limi ts___ ~__ 2 ~ ~

ParameterLevel

All measurement systems may produce wild data points. These points may be causedby temporary or intermittent malfunctions of the measurement system, or they mayrepresent actual variations in the measurement. Errors of this type cannot be estimated aspart of the uncertainty of the measurement. The points are out-of-control points for thesystem and are meaninglessas steady-state test data.They should be discarded.Figure D-l shows two spurious data points (sometimescalled outliers).

Fig. D-1 Outliers Outside the Range ofAcceptable Data

All data should beinspected for wild datapoints as a continuing qualitycontrol check on the measurement process. Identification criteria should be basedon engineering analysis of instrumentation, thermodynamics, flow profiles, and pasthistory with similar data. To ease the burden of scanning large masses of data, acomputerized routine is available to scan steady-state data and flag suspected outliers.The flagged points should then be subjected to a comprehensive engineering analysis.

This routine is intended to be used in scanning small samples of data from a largenumber of parameters at many time slices. The work of paging through volumes of datacan be reduced to a manageable job with this approach. The computer will scan the dataand flag suspect points. The engineer, relieved of the burden of scanning the data, canclosely examine each suspected wild point.

ARNOLD ENGINEERING DEVELOPMENT CENTER OUTLIER METHOD

Several general purpose outlier techniques were reviewed and discussed in the ICRPGHandbook CPIA 180. The U.S. Air Force Arnold Engineering Development Center(AEDC) has developed a new technique to flag outliers in small or moderately sizedsamples of data. This technique was compared with the Thompson's Tau Technique usedin ICRPG CPIA 180. The AEDC method as compared with the Thompson's Tau methoddetects a larger proportion of the outliers in the data, and when no outliers are present,it flags fewer good points. The AEDC method is useful for computer routines since it isfast and requires little core storage. The method discriminates between good data andoutliers by examining how far each point lies from the average value.

The first step is to calculate an average value (X) and a standard deviation (s) fromthe data.

163

AEDC-TR-73.5

xN X.~~i=l N

s = ±

N~ (X. _ X)2i=l 1

N-1

Then, from the sample size (N), a test value (C) is calculated from

c = -1.6819236 + 1.6386898N - 0.00721312N 2

21.0 + 0.5928677 2N - 0.00355709N

when N < 65. When N ~ 65, C = 3 is used. Each data point is tested to determine if itfalls in the interval, average value plus or minus the standard deviation times C, Le.,

-X ± Cs

If a data point falls outside the interval, it is flagged as an outlier.

The formula for calculating C was determined at AEDC and was based onengineering judgment to determine the expected intervals for good data in the range of10 to 30 samples. Then the curve (C) was fit to the data (Fig. D-2). For sample sizes of65 or greater, C = 3 should be used.

AN EXAMPLE OF THE AEDC METHOD

The AEDC Fortran Subroutine is used to test the data in Table D-I for outliers.

C

N

Fig. 0-2 Parameter C versus Sample Size

Table 0-1Pressure Data

for OutlierTest

Point Pressure,No. psia

1 12.962 13.153 13.014 13.115 13.306 13.687 13.268 13.109 12.84

10 13.1911 13.2512 13.3913 13.1114 13.0315 12.96

164

AEDC-TR-73-5

The first step is to calculate the mean (X) and standard deviation (s) for the data:

x

15~ X.i=1 1

1513.156

The test value (C) is calculated based on the sample size (N).

C -1.6819236 + 1.6386898N - 0.00721312N 2 2= = .3398

1.0 + 0.59286772N - 0.00355709N 2

Table D-2 lists the deviations (Xi - X) for the data. In thiscase, point number six has a deviation which is greater than0.4813. It is flagged as an outlier and printed out by thesubroutine.

Table 0-2

Point DeviationNo. (Xi - X)

1 -0.1962 -0.0063 -0.1464 -0.0465 0.1446 0.5247 0.1048 -0.0569 -0.316

10 0.03411 0.09412 0.23413 -0.04614 -0.12615 -0.196

±0.2057±s

Eacl!- data point is checked to determine if it falls in theinterval X±Cs, which in this case is 12.674~to 13.6373. Aconvenient method for doing this is to subtract X from each datapoint to determine the deviation (Xi - X). Then, this is checkedagainst Cs =2.3398 times 0.2057 =0.4813.

THE FORTRAN SUBROUTINE

The following is a listing of a subroutine to implement the AEDC outlier rejectionmethod in a computer program. The subroutine is written in American National StandardFortran IV. The subroutine was compiled and run on an IBM 360, model 75, with aG-ievel compiler operating under IBM release 20.1.

SUBROUTINE CAVG ( XBAR. SIG • X • N 9 IELIMcccc·ccccccccccccc

XAAR

x

SIG

N

IELIM

AEDC OUTLIER REJECTION SUBROUTINE

MEAN CALCULATED AFTER OUTLIERS HAVE REEN REMOVED FROMDATA

INPUT ARRAY OF SAMPLE DATA

STANDARD DEVIATION CALCULATED AFTER OUTLIERS HAVE BEENREMOVED FROM THE DATA

NUMBER OF DATA POINTS IN DATA SAMPLE. X

NUMAER OF OUTLIERS REJECTED IN SAMPLE OF DATA

165

AEDC·TR·73-5

C IF A SAMPLE DATA POINT IS FOUND TO BE AN OUTLIER, ITS VALUE J$C SET TO 0.0 •CC SA~PLE DATA POINTS EQUAL TO ZERO INPUT TO THE SUBROUTINE AREC DISCARDED FROM THE SAMPLE.CC

103

999

100

-

.101

DIMENSION Xl 1) • DEll 1.00)INO IB 0IELIM III aWRITEC6,10Z)FORMATe/' BEFORE REMOVING OuTLIERS'/)SUM III O.NN III 0DO :3 III1,NIF e x( I) • EQ. o. ) GO TO :3NN III NN -+ 1SUM : SUM -+ Xe I )

3 CONTINUEIF ( NN .LT. Z ) RETURNXBAR = SUM I NNSUMZ = o.DO 4 I == l,NIF C X(I) • EQ. O. ) GO TO 4+DEleI) III ABS( XCI) - XBAR )SUM2 III SUMZ -+ DElCI)**Z

4 CONTINUEC == 3.SIG III SQRT C SUM2 I e NN - 1 »wRITEe6,101)XBAR.SIG,NNFORMATe/' MEAN==',EZO.7,5X,'STD. OEV.III',EZO.7.5X,'N=',IS/)IF ( INO .EQ. 1 ) RETURNIF C NN .lE. 65 ) C B (-1.681923600 + l.6:386B9800 * NN

1 -.0072131200 * NN * NN ) I C 1.002 +.5929617200 * NN - .0035570900 * NN * NN )

CS B C * SIGK = NNDO 5 I == I, NIF ( XCI) .EQ. O••OR. OEl(I) .lE. CS ) GO TO 5NN B NN - 1WRITEl6,lOO)I.X(IJFORMAT(/5X,'POINT NO.',I5,' VAlUEIII',E20.7,' HAS BEEN REJECTEO'/)XCI) :: O.

5 CONTINUEIf-lIM:: K - NNIND B 1IF l K .EQ. NN) RETURNWRITE(6,103)FORMAT(/' AFTER REMOVING OUTlIERS'/)GO TO 1CONTINUEEND

1021

166

APPENDIX ETABLES

This section of the Appendix presents the tables of the Student's "t" distribution,the F table for comparison of precision indices, and the Thompson's Tau table for theoutlier test.

STUDENT'S Ut" TABLE

The table of Student's "t" distribution (Table E-l) presents the two-tailed9S-percent "t" values for the degrees of freedom from one to 30. Above 30, round thevalue to 2.0.

Table E-1 Two-Tailed Student's Ut" Table

t

Degrees of Degrees ofFreedom "t" Freedom "t"

1 12.706 17 2.1102 4.303 18 2.1013 3.182 19 2.0934 2.776 20 2.0865 2.571 21 2.0806 2.447 22 2.0747 2.365 23 2.0698 2.306 24 2.0649 2.262 25 2.060

10 2.228 26 2.05611 2.201 27 2.05212 2.179 28 2.04813 2.160 29 2.04514 2.145 30 2.04215 2.13116 2.120 31 or more use 2.0

The table is used to provide an interval estimate of the true value about an observedvalue. The interval is the measurement. plus and minus the standard deviation of theobserved value times the "t" value (for the degrees of freedom of that standarddeviation) :

interval = measurement ± t9S

S

167

AEDC-TR-73-5

The 95-percent Student's "t" value for a standard deviation of 50 Ib with 17 deg offreedom is 2.110. The interval is

F TABLE

measurement ± 2.11 x 50 measurement ± 105.50 Ib

The table of 95-percent F values (Table E-2) is presented for tests for a significantincrease in precision index. The test is performed by dividing the square of the newprecision index by the square of the old index:

F calculated

Snew

2

This calculated value is compared with the F table value for f1 and f2 degrees offreedom; f1 is the degrees of freedom for Snew 2, and f2 is the degrees of freedom forSold 2.

For example, suppose that the pooled precision index for a force measuring device is0.05 percent based on four sets of 5 tests each (total of 16 degrees of freedom). A newestimate of precision index is 0.10 percent based on a sample size of 5. The F calculatedvalue is:

F calculated

Snew

20.01000.002.E::

4.00

f1 = 4 and f2 = 16, the table value of F is 3.01. Because 4.00 is greater than 3.01, thisindicates the new index is significantly larger than the old index.

THOMPSON'S TAU TABLE

Thompson's Tau Table (Table E-3) is presented to aid in detecting outliers or baddata in a sample. The table is given for several levels of significance (p). If p = 0.05 levelis used, this sets the probability of a good point at 5 percent. The procedure for the testis this:

1. From the data (Xj; i = 1,n) containing the suspected outlier X', twostatistics are calculated:

x

N~ X.i=l 1

N

168

AE DC-T R-73-5

Table E-2 5-percent Fractiles of the F Distribution

S2

F(f1

, f 2)new

=-2-

Degrees of Freedom for the Numerator (f1) Sold

1 2 3 4 5 6 7 8 9 10 12 15 17

1 161 200 216 225 230 234 237 239 241 242 244 246 2472 18.5 19.0 19.2 19.2 19.3 19.3 19.4 19.4 19.4 19.4 19.4 19.4 19.43 10.1 9.55 9.28 9.12 9.01 8.94 8.89 8.85 8.81 8.79 8.74 8.70 8.684 7.71 6.94 6.59 6.39 6.26 6.16 6.09 6.04 6.00 5.96 5.91 5.86 5.835 6.61 5.79 5.41 5.19 5.05 4.95 4.88 4.82 4.77 4.74 4.68 4.62 4.596 5.99 5.14 4.76 4.53 4.39 4.28 4.21 4.15 4.10 4.06 4.00 3.94 3.917 5.59 4.74 4.35 4.12 3.97 3.87 3.79 3.73 3.68 3.64 3.57 3.51 3.488 5.32 4.46 4.07 3.84 3.69 3.58 3.50 3.44 3.39 3.35 3.28 3.22 3.199 5.12 4.26 3.86 3.63 3.48 3.37 3.29 3.23 3.18 3.14 3.07 3.01 2.97

10 4.96 4.10 3.71 3.48 3.33 3.22 3.14 3.07 3.02 2.98 2.91 2.85 2.81

,,-.... 11 4.84 3.98 3.59 3.36 3.20 3.09 3.01 2.95 2.90 2.85 2.79 2.72 2.69~ 12 4.75 3.89 3.49 3.26 3.11 3.00 2.91 2.85 2.80 2.75 2.69 2.62 2.58'-" 13 4.67 3.81 3.41 3.18 3.03 2.92 2.83 2.77 2.71 2.67 2.60 2.53 2.50l-l 14 4.60 3.74 3.34 3.11 2.96 2.85 2.76 2.70 2.65 2.60 2.53 2.46 2.430

+oJ 15 4.54 3.68 3.29 3.06 2.90 2.79 2.71 2.64 2.59 2.54 2.48 2.40 2.37CdI=: 16 4.49 3.63 3.24 3.01 2.85 2.74 2.66 2.59 2.54 2.49 2.42 2.35 2.32.~

S 17 4.45 3.59 3.20 2.96 2.81 2.70 2.61 2.55 2.49 2.45 2.38 2.31 2.270I=: 18 4.41 3.55 3.16 2.93 2.77 2.66 2.58 2.51 2.46 2.41 2.34 2.27 2.23<U

0 19 4.38 3.52 3.13 2.90 2.74 2.63 2.54 2.48 2.42 2.38 2.31 2.23 2.20<U 20 4.35 3.49 3.10 2.87 2.71 2.60 2.51 2.45 2.39 2.35 2.28 2.20 2.17..c:

+oJ

l-l 21 4.32 3.47 3.07 2.84 2.68 2.57 2.49 2.42 2.37 2.32 2.25 2.18 2.140 22 4.30 3.44 1.05 2.82 2.66 2.55 2.46 2.40 2.34 2.30 2.23 2.15 2.11

lH

S 23 4.28 3.42 3.03 2.80 2.64 2.53 2.44 2.37 2.32 2.27 2.20 2.13 2.090 24 4.26 3.40 3.01 2.78 2.62 2.51 2.42 2.36 2.30 2.25 2.18 2.11 2.07

"'c:l<U 25 4.24 3.39 2.99 2.76 2.60 2.49 2.40 2.34 2.28 2.24 2.16 2.09 2.05(1)l-l 26 4.23 3.37 2.98 2.74 2.59 2.47 2.39 2.32 2.27 2.22 2.15 2.07 2.03J:x.I

lH 28 4.20 3.34 2.95 2.71 2.56 2.45 2.36 2.29 2.24 2.19 2.12 2.04 2.000 30 4.17 ,3.32 2.92 2.69 2.53 2.42 2.33 2.27 2.21 2.16 2.09 2.01 1. 98Ul 34 4.13 3.28 2.88 2.65 2.49 2.38 2.29 2.23 2.17 2.12 2.05 1.97 1. 93<U<U 38 4.io 3.24 2.85 2.62 2.46 2.35 2.26 2.19 2.14 2.09 2.02 1.94 1.90l-lb1)<U 40 4.08 3.23 2.84 2.61 2.45 2.34 2.25 2.18 2.12 2.08 2.00 1. 92 1. 890

44 4.06 3.21 2.82 2.58 2.43 2.31 2.23 2.16 2.10 2.05 1.98 1.90 1.8648 4.04 3.19 2.80 2.57 2.41 2.29 2.21 2.14 2.08 2.03 1.96 1.88 1.8450 4.03 3.18 2.79 2.56 2.40 2.29 2.20 2.13 2.07 2.03 1.95 1.87 1.8355 4.02 3.16 2.77 2.54 2.38 2.27 2.18 2.11 2.06 2.01 1.93 1.85 1.8160 4.00 3.15 2.76 2.53 2.37 2.25 2.17 2.10 2.04 1. 99 1.92 1.84 1.8065 3.99 3.14 2.75 2.51 2.36 2.24 2.15 2.08 2.03 1. 98 1.90 1.82 1. 7880 3.96 3.11 2.72 2.49 2.33 2.21 2.13 2.06 2.00 1.95 1.88 1.79 1.75

100 3.94 3.09 2.70 2.46 2.31 2.19 2.10 2.03 1. 97 1. 93 1. 85 1.77 1. 73150 3.90 3.06 2.66 2.43 2.27 2.16 2.07 2.00 1.94 1. 89 1.82 1. 73 1.69

200 3.89 3.04 2.65 2.42 2.26 2.14 2.06 1.98 1.93 1. 88 1. 80 1. 72 1. 671000 3.85 3.00 2.61 2.38 2.22 2.11 2.02 1.95 1.89 i.84 1.76 1.68 1.63

co 3.84 3.00 2.60 2.37 2.21 2.10 2.01 1.94 1.88 1.83 1.75 1.67 1.62

169

AE DC·T R-73·5

Table E-2 (Concluded)

s2

F(f l , f2

)new

= 2.1.

SoldDegrees of Freedom for the Numerator (f1)

19 20 22 24 26 30 35 40 50 100 500 co

248 248 249 249 249 250 251 251 252 253 254 254 119.4 19.4 19.5 19.5 19.5 19.5 19.5 19~5 19.5 19'.5 19.5 19.5 28.67 8.66 8.65 8.64 8.63 8.62 8.60 8.59 8.58 8.55 8.53 8.53 35.81 5.80 5.79 5.77 5.76 5.75 5.73 5.72 5.70 5.66 5.64 5.63 44.57 4.56 4.54 4.53 4.52 4.50 4.48 4.46 4.44 4.41 4.37 4.37 5J.88 3.87 3.86 3.84 3.83 3.81 3.79 3.77 3.75 3.71 3.68 3.67 63.46 3.44 3.43 3.41 3.40 3.38 3.36 3.34 3.32 3.27 3.24 3.23 73.16 3.15 3.13 3.12 3.10 3.08 3.06 3.04 3.02 2.97 2.94 2.93 82.95 2.94 2.92 2.90 2.89 2.86 2.84 2.83 2.80 2.76 2.72 2.71 92.78 2.77 2.75 2.74 2.72 2.70 2.68 2.66 2.64 2.59 2.55 2.54 10

2.66 2.65 2.63 2.61 2.59 2.57 2.55 2..53 2.51 2.46 2.42 2.40 112.56 2.54 2.52 2.51 2.49 2.47 2.44 2.43 2.40 2.35 2.31 2.30 12 t::l

2.47 2.46 2.44 2.42 2.41 2.38 2.36 2.34 2.31 2.26 2.22 2.21 13 ro(JQ

2.40 2.39 2.37 2.35 2.33 2.31 2.28 2.27 2.24 2.19 2.14 2.13 14 1'1(1)

2.34 2.33 2.31 2.29 2.27 2.25 2.22 2.20 2.18 2.12 2.08 2.07 15(1)CIJ

2.29 2.28 2.25 2.24 2.22 2.19 2.17 2.15 2.12 2.07 2.02 2.01 16 0

2.24 2.23 2.21 2.19 2.17 2.15 2.12 2.10 2.08 2.02 1.97 1.96 17 H1

2.20 2.19 2.17 2.15 2.13 2.11 2.08 2.06 2.04 1.98 1.93 1.92 18i"%j1'1

2.17 2.16 2.13 2.11 2.10 2.07 2.05 2.03 2.00 1.94 1.89 1.88 19(1)(1)

.2.14 2.12 2.10 2.08 2.07 2.04 2.01 1.99 1.97 1.91 1.86 1.84 20p..0S

2.11 2.10 2.07 2.05 2.04 2.01 1.98 1.96 1.94 1.88 1.82 1.81 21 H1

2.08 2.07 2.05 2.03 2.01 1.98 1.96 1.94 1.91 1.85 1.80 1. 78 2201'1

2.06 2.05 2.02 2.00 1.99 1.96 1.93 1.91 1.88 1.82 1. 77 1.76 23 t::l

2.04 2.03 2.00 1.98 1.97 1.94 1.91 1.89 1.86 1.80 1.75 1.73 24(1)

::s2.02 2.01 . 1.98 1.96 1.95 1.92 1.89 1.87 1.84 1. 78 1.73 1.71 25 0s2.00 1. 99 1.97 1.95 1.93 1.90 1.87 1.85 1.82 1.76 1. 71 1.69 26

/-'-::s

1. 97 1. 96 1.93 1.91 1.90 1.87 1. 84 1. 82 1.79 1. 73 1. 67 1.65 28 IUrt

1. 95 1.93 1.91 1~89 1.87 1.84 1. 81 1. 79 1. 76 1. 70 1.64 1.62 30 01'1

1. 90 1.89 1.86 1.84 1.82 1.80 1.77 1.75 1. 71 1.65 1.59 1.57 34 r--.

1. 87 1.85 1.83 1.81 1.79 1.76 1. 73 1. 71 1.68 1.61 1.54 1.53 38 H1N'-'

1. 85 1.84 1.81 1.79 1.77 1.74 1.72 1.69 1.66 1.59 1.53 1.51 40.1. 83 1.81 1.79 1.77 1.75 1.72 1.69 1.67 1.63 1.56 1.49 1.48 441.8~ 1.79 1.77 1.75 1.73 1.70 1.67 1.64 1.61 1.54 1.47 1.45 481.80 1. 78 1.76 1.74 1.72 1.69 1.66 1.63 1.60 1.52 1.46 1.44 501. 78 1. 76 1.74 1.72 1.70 1.67 1.64 1.61 1.58 1.50 1.43 1.41 551.76 1.75 . ,1.72 1.70 1.68 1.65 1.62 1.59 1.56 1.48 1.41 1.39 601.75 1.7·3 1.71 1.69 1.67 1.63 1.60 1.58 1.54 1.46 1.39 1.37 651. 72 1. 70 1.'68 1. 65 1.63 1.60 1.57 1.54 1.51 1.43 1.35 1.32 801.69 1.68 1.65 1.63 1.61 1.57 1.54 1.52 1.48 1.39 1.31 1.28 1001.66 1.64 1.61 1.59 1.57 1.53 1.50 1.48 1.44 1.34 1.25 1.22 150

1.64 1. 62 1.60 1.57 1.55 1.52 1.48 1.46 1.41 1.32 1.22 1.19 2001.60 1.58 1.55 1.53 1.51 1.47 1.44 1.41 1. 36 1. 26 1.13 1.08 10001.59 1.57 1.54 1.52 1.50 1.46 1.42 1.39 1.35 1.24 1.11 1. 00 co

170

SampleSize

Table E-3 Thompson's Tau

Level of Significance

AEDC-TR-73-5

N p= .1 .05 .02 .01

3 1.3968 1.4099 1.41352 1.4140394 1.559 1.6080 1.6974 1.71475 1.611 1.757 1.869 1.9175

6 1.631 1.814 1.973 2.05097 1.640 1.848 2.040 2.1428 1.644 1.870 2.087 2.2079 1.647 1.885 2.121 2.256

10 1.648 1.895 2.146 2.294

11 1.648 1.904 2.166 2.32412 1.649 1.910 2.183 2.34813 1.649 1.915 2.196 2.36814 1.649 1.919 2.207 2.38515 1.649 1.923 2.216 2.399

16 1.649 1.926 2.224 2.41117 1.649 1.928 2.231 2.42218 1.649 1.931 2.237 2.43219 1.649 1.932 2.242 2.44020 1.649 1.934 2.247 2.447

21 1.649 1.936 2.251 2.45422 1.649 1. 937 2.255 2.46023 1.649 1. 938 2.259 2.46524 1.649 1.940 2.262 2.47025 1.649 1. 941 2.264 -2.475

26 1.648 1.942 2;267 2.47927 1.648 1.942 2.269 2.483.28 1.648 1.943 2.272 2.48729 1.648 1.944 2.274 2.49030 1.648 1. 944 2.275 2.493

31 1.648 1. 945 2.277 2.49532 1.648 1.945 2.279 2.498

CD 1.64485 1.95996 2.32634 2.57582

171

AED C·TR·73·5

and

SD

N _ 2L (x. - X)i=l 1

N

These are the average value and standard deviation of the sample.

2. Calculate the difference in (absolute value) between the average value Xand the outlier X ':

3. Entering the table for Nand P, (take P = 0.05) read a value of Tau.

4. The comparison is made between 0 and the product of SD and Tau. If 0is larger or equal to that product, the data point is declared an outlier. Anew mean and standard deviation must be calculated. If 0 is smaller, thepoint is not rejected.

Example:

In the following sample of 15 data points, X and S were calculated to be 9.949 and0.997:

9.55811.44710.472

10.47811.48510.310

9.609 9.58211.067 9.1 737.416 9.488

'L-Suspected outlier

9.58310.303 .9.257

The test is to compare 0 (the absolute difference between the suspected outlier andthe average value) with the product Tau value times calculated standard deviation.

o = lx-x'/ = /9.949-7.4161 = 2.533

Tau X SD = 1.923 x 0.997 = 1.917

where Tau is the Table E-3 value for P = 0.05 and N =15. Since 0 is greater, the point7.416 is discarded.

172

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l. ORIGINATING ACTIVITY (Corporate aut,!or) 2a. REPORT SECURITY CLASSIFICATION

Arnold Engineering Development Center,Arnold Air Force Station, Tennessee 37389

3. REPORT TITLE

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UNCLASSIFIED

N/A

HANDBOOK, UNCERTAINTY IN GAS TURBINE MEASUREMENTS

4. DESCRIPTIVE NOTES (Type 01 report and Inclusive dates)

Final Report~. AU THOR(S) (First name, middle Initial, last name)

Dr. R. B. Abernethy et al., Pratt & Whitney AircraftJ. W Thompson, Jr., ARO, Inc.

6. REPORT DATE

Eebruary 19738a. CON TRAC T OR GRAN T NO.

b. PROJECT NO.

c. Program Element 65802F

d.

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AEDC-TR-73-5

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ARO-ETF-TR-72-60


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13. ABSTRACT

12. SPONSORING MILITARY ACTIVITY

Arnold Engineering Development CenterArnold Air Force StationTennessee 373'~9

The lack of a standard method for estimating the errors associatedwith gas turbine performance data has made it impossible to comparemeasurement systems between facilities, and there has been confusionover the interpretation of error analysis. Therefore, a standard uncertainty methodology is proposed in this Handbook. The mathematicaluncertainty model presented is based on two components of measurementerror: the fixed (bias) error and the random (precision) error. Theresult of applying the model is an estimate of the error in the measuredperformance parameter. The uncertainty estimate is the interval aboutthe measurement which is expected to encompass the true value. Thepropagation of error from basic measurements through calculated performance paramet~rs is presented, Traceability of measurement back to theNational Bu~eau of Standards and associated error sources is reviewed.

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handbook

gas turbine engines

measurement

uncertainty

error analysis

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